diff --git a/ci/code_checks.sh b/ci/code_checks.sh
index 9453edb0c88d8..c74b2b44cf346 100755
--- a/ci/code_checks.sh
+++ b/ci/code_checks.sh
@@ -241,6 +241,7 @@ if [[ -z "$CHECK" || "$CHECK" == "docstrings" ]]; then
         pandas.api.extensions.ExtensionArray.factorize \
         pandas.api.extensions.ExtensionArray.fillna \
         pandas.api.extensions.ExtensionArray.insert \
+        pandas.api.extensions.ExtensionArray.interpolate \
         pandas.api.extensions.ExtensionArray.isin \
         pandas.api.extensions.ExtensionArray.isna \
         pandas.api.extensions.ExtensionArray.ravel \
diff --git a/doc/source/reference/extensions.rst b/doc/source/reference/extensions.rst
index b33efd388bd60..63eacc3f6d1d9 100644
--- a/doc/source/reference/extensions.rst
+++ b/doc/source/reference/extensions.rst
@@ -51,6 +51,7 @@ objects.
       api.extensions.ExtensionArray.factorize
       api.extensions.ExtensionArray.fillna
       api.extensions.ExtensionArray.insert
+      api.extensions.ExtensionArray.interpolate
       api.extensions.ExtensionArray.isin
       api.extensions.ExtensionArray.isna
       api.extensions.ExtensionArray.ravel
diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index 517baa648d805..50e8bb0087d73 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -111,6 +111,7 @@ Other enhancements
 - :meth:`Series.str.join` now supports ``ArrowDtype(pa.string())`` (:issue:`53646`)
 - :meth:`SeriesGroupby.agg` and :meth:`DataFrameGroupby.agg` now support passing in multiple functions for ``engine="numba"`` (:issue:`53486`)
 - :meth:`SeriesGroupby.transform` and :meth:`DataFrameGroupby.transform` now support passing in a string as the function for ``engine="numba"`` (:issue:`53579`)
+- Added :meth:`ExtensionArray.interpolate` used by :meth:`Series.interpolate` and :meth:`DataFrame.interpolate` (:issue:`53659`)
 - Added ``engine_kwargs`` parameter to :meth:`DataFrame.to_excel` (:issue:`53220`)
 - Added a new parameter ``by_row`` to :meth:`Series.apply`. When set to ``False`` the supplied callables will always operate on the whole Series (:issue:`53400`).
 - Groupby aggregations (such as :meth:`DataFrameGroupby.sum`) now can preserve the dtype of the input instead of casting to ``float64`` (:issue:`44952`)
@@ -118,6 +119,7 @@ Other enhancements
 - Many read/to_* functions, such as :meth:`DataFrame.to_pickle` and :func:`read_csv`, support forwarding compression arguments to lzma.LZMAFile (:issue:`52979`)
 - Performance improvement in :func:`concat` with homogeneous ``np.float64`` or ``np.float32`` dtypes (:issue:`52685`)
 - Performance improvement in :meth:`DataFrame.filter` when ``items`` is given (:issue:`52941`)
+-
 
 .. ---------------------------------------------------------------------------
 .. _whatsnew_210.notable_bug_fixes:
diff --git a/pandas/_typing.py b/pandas/_typing.py
index 9d4acbe76ba15..ffe9e6b319dfd 100644
--- a/pandas/_typing.py
+++ b/pandas/_typing.py
@@ -307,6 +307,26 @@ def closed(self) -> bool:
 
 # Arguments for fillna()
 FillnaOptions = Literal["backfill", "bfill", "ffill", "pad"]
+InterpolateOptions = Literal[
+    "linear",
+    "time",
+    "index",
+    "values",
+    "nearest",
+    "zero",
+    "slinear",
+    "quadratic",
+    "cubic",
+    "barycentric",
+    "polynomial",
+    "krogh",
+    "piecewise_polynomial",
+    "spline",
+    "pchip",
+    "akima",
+    "cubicspline",
+    "from_derivatives",
+]
 
 # internals
 Manager = Union[
diff --git a/pandas/core/arrays/base.py b/pandas/core/arrays/base.py
index 78ecd93d5cc75..ceac8e22426d9 100644
--- a/pandas/core/arrays/base.py
+++ b/pandas/core/arrays/base.py
@@ -78,6 +78,7 @@
         AxisInt,
         Dtype,
         FillnaOptions,
+        InterpolateOptions,
         NumpySorter,
         NumpyValueArrayLike,
         PositionalIndexer,
@@ -90,6 +91,8 @@
         npt,
     )
 
+    from pandas import Index
+
 _extension_array_shared_docs: dict[str, str] = {}
 
 
@@ -118,6 +121,7 @@ class ExtensionArray:
     fillna
     equals
     insert
+    interpolate
     isin
     isna
     ravel
@@ -155,6 +159,7 @@ class ExtensionArray:
     * take
     * copy
     * _concat_same_type
+    * interpolate
 
     A default repr displaying the type, (truncated) data, length,
     and dtype is provided. It can be customized or replaced by
@@ -753,6 +758,27 @@ def argmax(self, skipna: bool = True) -> int:
             raise NotImplementedError
         return nargminmax(self, "argmax")
 
+    def interpolate(
+        self,
+        *,
+        method: InterpolateOptions,
+        axis: int,
+        index: Index,
+        limit,
+        limit_direction,
+        limit_area,
+        fill_value,
+        copy: bool,
+        **kwargs,
+    ) -> Self:
+        """
+        See DataFrame.interpolate.__doc__.
+        """
+        # NB: we return type(self) even if copy=False
+        raise NotImplementedError(
+            f"{type(self).__name__} does not implement interpolate"
+        )
+
     def fillna(
         self,
         value: object | ArrayLike | None = None,
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index f13a5e8b560a4..bef8b55b876a2 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -58,6 +58,7 @@
     Dtype,
     DtypeObj,
     F,
+    InterpolateOptions,
     NpDtype,
     PositionalIndexer2D,
     PositionalIndexerTuple,
@@ -2233,23 +2234,23 @@ def copy(self, order: str = "C") -> Self:
     def interpolate(
         self,
         *,
-        method,
+        method: InterpolateOptions,
         axis: int,
         index: Index,
         limit,
         limit_direction,
         limit_area,
-        inplace: bool,
+        copy: bool,
         **kwargs,
     ) -> Self:
         """
         See NDFrame.interpolate.__doc__.
         """
-        # NB: we return type(self) even if inplace=True
+        # NB: we return type(self) even if copy=False
         if method != "linear":
             raise NotImplementedError
 
-        if inplace:
+        if not copy:
             out_data = self._ndarray
         else:
             out_data = self._ndarray.copy()
@@ -2264,7 +2265,7 @@ def interpolate(
             limit_area=limit_area,
             **kwargs,
         )
-        if inplace:
+        if not copy:
             return self
         return type(self)._simple_new(out_data, dtype=self.dtype)
 
diff --git a/pandas/core/arrays/numpy_.py b/pandas/core/arrays/numpy_.py
index 8b6e49ccf2d41..990b55cb40366 100644
--- a/pandas/core/arrays/numpy_.py
+++ b/pandas/core/arrays/numpy_.py
@@ -36,6 +36,7 @@
         AxisInt,
         Dtype,
         FillnaOptions,
+        InterpolateOptions,
         NpDtype,
         Scalar,
         Self,
@@ -261,20 +262,20 @@ def pad_or_backfill(
     def interpolate(
         self,
         *,
-        method,
+        method: InterpolateOptions,
         axis: int,
         index: Index,
         limit,
         limit_direction,
         limit_area,
-        inplace: bool,
+        copy: bool,
         **kwargs,
     ) -> Self:
         """
         See NDFrame.interpolate.__doc__.
         """
-        # NB: we return type(self) even if inplace=True
-        if inplace:
+        # NB: we return type(self) even if copy=False
+        if not copy:
             out_data = self._ndarray
         else:
             out_data = self._ndarray.copy()
@@ -290,7 +291,7 @@ def interpolate(
             limit_area=limit_area,
             **kwargs,
         )
-        if inplace:
+        if not copy:
             return self
         return type(self)._simple_new(out_data, dtype=self.dtype)
 
diff --git a/pandas/core/generic.py b/pandas/core/generic.py
index a4b4466e8d609..b9cf5fca30b2f 100644
--- a/pandas/core/generic.py
+++ b/pandas/core/generic.py
@@ -62,6 +62,7 @@
     IgnoreRaise,
     IndexKeyFunc,
     IndexLabel,
+    InterpolateOptions,
     IntervalClosedType,
     JSONSerializable,
     Level,
@@ -7753,27 +7754,7 @@ def replace(
     @final
     def interpolate(
         self,
-        method: Literal[
-            "linear",
-            "time",
-            "index",
-            "values",
-            "pad",
-            "nearest",
-            "zero",
-            "slinear",
-            "quadratic",
-            "cubic",
-            "barycentric",
-            "polynomial",
-            "krogh",
-            "piecewise_polynomial",
-            "spline",
-            "pchip",
-            "akima",
-            "cubicspline",
-            "from_derivatives",
-        ] = "linear",
+        method: InterpolateOptions = "linear",
         *,
         axis: Axis = 0,
         limit: int | None = None,
diff --git a/pandas/core/internals/blocks.py b/pandas/core/internals/blocks.py
index 832e1560ae55e..eec94a2d61c26 100644
--- a/pandas/core/internals/blocks.py
+++ b/pandas/core/internals/blocks.py
@@ -35,6 +35,7 @@
     F,
     FillnaOptions,
     IgnoreRaise,
+    InterpolateOptions,
     QuantileInterpolation,
     Self,
     Shape,
@@ -600,13 +601,13 @@ def _get_values_and_refs(self, using_cow, inplace):
     @final
     def _get_refs_and_copy(self, using_cow: bool, inplace: bool):
         refs = None
-        arr_inplace = inplace
+        copy = not inplace
         if inplace:
             if using_cow and self.refs.has_reference():
-                arr_inplace = False
+                copy = True
             else:
                 refs = self.refs
-        return arr_inplace, refs
+        return copy, refs
 
     # ---------------------------------------------------------------------
     # Replace
@@ -1383,7 +1384,7 @@ def pad_or_backfill(
     def interpolate(
         self,
         *,
-        method: FillnaOptions = "pad",
+        method: FillnaOptions | InterpolateOptions = "pad",
         axis: AxisInt = 0,
         index: Index | None = None,
         inplace: bool = False,
@@ -1402,17 +1403,8 @@ def interpolate(
                 return [self.copy(deep=False)]
             return [self] if inplace else [self.copy()]
 
-        try:
-            m = missing.clean_fill_method(method)
-        except ValueError:
-            m = None
-            # error: Non-overlapping equality check (left operand type:
-            # "Literal['backfill', 'bfill', 'ffill', 'pad']", right
-            # operand type: "Literal['asfreq']")
-            if method == "asfreq":  # type: ignore[comparison-overlap]
-                # clean_fill_method used to allow this
-                raise
-        if m is None and self.dtype == _dtype_obj:
+        # TODO(3.0): this case will not be reachable once GH#53638 is enforced
+        if not _interp_method_is_pad_or_backfill(method) and self.dtype == _dtype_obj:
             # only deal with floats
             # bc we already checked that can_hold_na, we don't have int dtype here
             # test_interp_basic checks that we make a copy here
@@ -1435,32 +1427,32 @@ def interpolate(
                 **kwargs,
             )
 
-        arr_inplace, refs = self._get_refs_and_copy(using_cow, inplace)
+        copy, refs = self._get_refs_and_copy(using_cow, inplace)
 
         # Dispatch to the PandasArray method.
         # We know self.array_values is a PandasArray bc EABlock overrides
-        if m is not None:
+        if _interp_method_is_pad_or_backfill(method):
             # TODO: warn about ignored kwargs, limit_direction, index...?
             new_values = cast(PandasArray, self.array_values).pad_or_backfill(
-                method=method,
+                method=cast(FillnaOptions, method),
                 axis=axis,
                 limit=limit,
                 limit_area=limit_area,
-                copy=not arr_inplace,
+                copy=copy,
             )
         else:
             assert index is not None  # for mypy
             new_values = cast(PandasArray, self.array_values).interpolate(
-                method=method,
+                method=cast(InterpolateOptions, method),
                 axis=axis,
                 index=index,
                 limit=limit,
                 limit_direction=limit_direction,
                 limit_area=limit_area,
-                inplace=arr_inplace,
+                copy=copy,
                 **kwargs,
             )
-        data = new_values._ndarray
+        data = extract_array(new_values, extract_numpy=True)
 
         nb = self.make_block_same_class(data, refs=refs)
         return nb._maybe_downcast([nb], downcast, using_cow)
@@ -1889,7 +1881,8 @@ def values_for_json(self) -> np.ndarray:
     def interpolate(
         self,
         *,
-        method: FillnaOptions = "pad",
+        method: FillnaOptions | InterpolateOptions = "pad",
+        index: Index | None = None,
         axis: int = 0,
         inplace: bool = False,
         limit: int | None = None,
@@ -1898,11 +1891,28 @@ def interpolate(
         **kwargs,
     ):
         values = self.values
-        if values.ndim == 2 and axis == 0:
-            # NDArrayBackedExtensionArray.fillna assumes axis=1
-            new_values = values.T.fillna(value=fill_value, method=method, limit=limit).T
+
+        if not _interp_method_is_pad_or_backfill(method):
+            imeth = cast(InterpolateOptions, method)
+            return super().interpolate(
+                method=imeth,
+                index=index,
+                axis=axis,
+                inplace=inplace,
+                limit=limit,
+                fill_value=fill_value,
+                using_cow=using_cow,
+                **kwargs,
+            )
         else:
-            new_values = values.fillna(value=fill_value, method=method, limit=limit)
+            meth = cast(FillnaOptions, method)
+            if values.ndim == 2 and axis == 0:
+                # NDArrayBackedExtensionArray.fillna assumes axis=1
+                new_values = values.T.fillna(
+                    value=fill_value, method=meth, limit=limit
+                ).T
+            else:
+                new_values = values.fillna(value=fill_value, method=meth, limit=limit)
         return self.make_block_same_class(new_values)
 
 
@@ -2245,45 +2255,6 @@ class DatetimeLikeBlock(NDArrayBackedExtensionBlock):
     def values_for_json(self) -> np.ndarray:
         return self.values._ndarray
 
-    def interpolate(
-        self,
-        *,
-        method: FillnaOptions = "pad",
-        index: Index | None = None,
-        axis: int = 0,
-        inplace: bool = False,
-        limit: int | None = None,
-        fill_value=None,
-        using_cow: bool = False,
-        **kwargs,
-    ):
-        values = self.values
-
-        # error: Non-overlapping equality check (left operand type:
-        # "Literal['backfill', 'bfill', 'ffill', 'pad']", right operand type:
-        # "Literal['linear']")  [comparison-overlap]
-        if method == "linear":  # type: ignore[comparison-overlap]
-            # TODO: GH#50950 implement for arbitrary EAs
-            arr_inplace, refs = self._get_refs_and_copy(using_cow, inplace)
-
-            new_values = self.values.interpolate(
-                method=method,
-                index=index,
-                axis=axis,
-                inplace=arr_inplace,
-                limit=limit,
-                fill_value=fill_value,
-                **kwargs,
-            )
-            return self.make_block_same_class(new_values, refs=refs)
-
-        elif values.ndim == 2 and axis == 0:
-            # NDArrayBackedExtensionArray.fillna assumes axis=1
-            new_values = values.T.fillna(value=fill_value, method=method, limit=limit).T
-        else:
-            new_values = values.fillna(value=fill_value, method=method, limit=limit)
-        return self.make_block_same_class(new_values)
-
 
 class DatetimeTZBlock(DatetimeLikeBlock):
     """implement a datetime64 block with a tz attribute"""
@@ -2597,3 +2568,14 @@ def external_values(values: ArrayLike) -> ArrayLike:
     # TODO(CoW) we should also mark our ExtensionArrays as read-only
 
     return values
+
+
+def _interp_method_is_pad_or_backfill(method: str) -> bool:
+    try:
+        m = missing.clean_fill_method(method)
+    except ValueError:
+        m = None
+        if method == "asfreq":
+            # clean_fill_method used to allow this
+            raise
+    return m is not None
diff --git a/pandas/core/resample.py b/pandas/core/resample.py
index 9566a2f113b36..ef6a91b811c4a 100644
--- a/pandas/core/resample.py
+++ b/pandas/core/resample.py
@@ -89,7 +89,7 @@
         AxisInt,
         Frequency,
         IndexLabel,
-        QuantileInterpolation,
+        InterpolateOptions,
         T,
         TimedeltaConvertibleTypes,
         TimeGrouperOrigin,
@@ -837,7 +837,7 @@ def fillna(self, method, limit: int | None = None):
 
     def interpolate(
         self,
-        method: QuantileInterpolation = "linear",
+        method: InterpolateOptions = "linear",
         *,
         axis: Axis = 0,
         limit: int | None = None,