diff --git a/array_api_strict/_elementwise_functions.py b/array_api_strict/_elementwise_functions.py
index 7dc6c5c..8cec86a 100644
--- a/array_api_strict/_elementwise_functions.py
+++ b/array_api_strict/_elementwise_functions.py
@@ -14,6 +14,7 @@
 from ._array_object import Array
 from ._flags import requires_api_version
 from ._creation_functions import asarray
+from ._data_type_functions import broadcast_to, iinfo
 
 from typing import Optional, Union
 
@@ -325,14 +326,51 @@ def clip(
     if min is not None and max is not None and np.any(min > max):
         raise ValueError("min must be less than or equal to max")
 
-    result = np.clip(x._array, min, max)
-    # Note: NumPy applies type promotion, but the standard specifies the
-    # return dtype should be the same as x
-    if result.dtype != x.dtype._np_dtype:
-        # TODO: I'm not completely sure this always gives the correct thing
-        # for integer dtypes. See https://github.com/numpy/numpy/issues/24976
-        result = result.astype(x.dtype._np_dtype)
-    return Array._new(result, device=x.device)
+    # np.clip does type promotion but the array API clip requires that the
+    # output have the same dtype as x. We do this instead of just downcasting
+    # the result of xp.clip() to handle some corner cases better (e.g.,
+    # avoiding uint64 -> float64 promotion).
+
+    # Note: cases where min or max overflow (integer) or round (float) in the
+    # wrong direction when downcasting to x.dtype are unspecified. This code
+    # just does whatever NumPy does when it downcasts in the assignment, but
+    # other behavior could be preferred, especially for integers. For example,
+    # this code produces:
+
+    # >>> clip(asarray(0, dtype=int8), asarray(128, dtype=int16), None)
+    # -128
+
+    # but an answer of 0 might be preferred. See
+    # https://github.com/numpy/numpy/issues/24976 for more discussion on this issue.
+
+    # At least handle the case of Python integers correctly (see
+    # https://github.com/numpy/numpy/pull/26892).
+    if type(min) is int and min <= iinfo(x.dtype).min:
+        min = None
+    if type(max) is int and max >= iinfo(x.dtype).max:
+        max = None
+
+    def _isscalar(a):
+        return isinstance(a, (int, float, type(None)))
+    min_shape = () if _isscalar(min) else min.shape
+    max_shape = () if _isscalar(max) else max.shape
+
+    result_shape = np.broadcast_shapes(x.shape, min_shape, max_shape)
+
+    out = asarray(broadcast_to(x, result_shape), copy=True)._array
+    device = x.device
+    x = x._array
+
+    if min is not None:
+        a = np.broadcast_to(np.asarray(min), result_shape)
+        ia = (out < a) | np.isnan(a)
+
+        out[ia] = a[ia]
+    if max is not None:
+        b = np.broadcast_to(np.asarray(max), result_shape)
+        ib = (out > b) | np.isnan(b)
+        out[ib] = b[ib]
+    return Array._new(out, device=device)
 
 def conj(x: Array, /) -> Array:
     """