diff --git a/src/cmap/_colormap.py b/src/cmap/_colormap.py
index 97a0272e4..b15d1ad8d 100644
--- a/src/cmap/_colormap.py
+++ b/src/cmap/_colormap.py
@@ -5,6 +5,7 @@
 import base64
 import warnings
 from collections.abc import Iterable, Sequence
+from copy import copy
 from functools import partial
 from numbers import Number
 from typing import TYPE_CHECKING, Any, Callable, NamedTuple, Union, cast, overload
@@ -306,7 +307,9 @@ def __init__(
             name = value if isinstance(value, str) else "custom colormap"
 
         self.interpolation = _norm_interp(interpolation)
-        stops._interpolation = self.interpolation
+        if stops._interpolation != self.interpolation:
+            # stops may be owned by the caller: don't write the mode into it
+            stops = stops._with_interpolation(self.interpolation)
         self.color_stops = stops
         self.name = name
         self.identifier = _make_identifier(identifier or name)
@@ -456,6 +459,7 @@ def with_extremes(
             self.color_stops,
             name=self.name,
             category=self.category,
+            interpolation=self.interpolation,
             bad=bad,
             under=under,
             over=over,
@@ -1175,6 +1179,16 @@ def _reverser(func: LutCallable, x: NDArray) -> NDArray:
     def _is_reversed_lut_func(self, f: Callable) -> TypeGuard[partial]:
         return isinstance(f, partial) and f.func is self._reverser
 
+    def _with_interpolation(self, interpolation: Interpolation) -> ColorStops:
+        """Return a copy of self, with a different interpolation mode.
+
+        A shallow copy: rebuilding from `_lut_func` would call the user's callable
+        a second time, which is neither cheap nor guaranteed to be repeatable.
+        """
+        new = copy(self)
+        new._interpolation = interpolation
+        return new
+
     def reversed(self) -> ColorStops:
         """Return a new ColorStops object with reversed colors."""
         if (lut_func := self._lut_func) is not None:
diff --git a/tests/test_colormap.py b/tests/test_colormap.py
index c17ad6f7a..e403c76bf 100644
--- a/tests/test_colormap.py
+++ b/tests/test_colormap.py
@@ -99,6 +99,24 @@ def test_colorstops_reversed_does_not_mutate_source() -> None:
     npt.assert_array_equal(np.asarray(stops), before)
 
 
+def test_construction_does_not_change_source_interpolation() -> None:
+    stops = Colormap(["red", "blue"], interpolation="nearest").color_stops
+    before = stops.to_lut(4).copy()
+
+    Colormap(stops, interpolation="linear")
+
+    npt.assert_array_equal(stops.to_lut(4), before)
+
+
+def test_with_extremes_preserves_interpolation() -> None:
+    cmap = Colormap(["red", "blue"], interpolation="nearest")
+
+    new = cmap.with_extremes(bad="red")
+
+    assert new.interpolation == "nearest"
+    npt.assert_array_equal(new.lut(4), cmap.lut(4))
+
+
 def test_colormap_copy() -> None:
     """Test Colormap copy."""
     import pickle