diff --git a/altair/utils/schemapi.py b/altair/utils/schemapi.py
index fdf0d6594..9bcf039c3 100644
--- a/altair/utils/schemapi.py
+++ b/altair/utils/schemapi.py
@@ -25,6 +25,7 @@
     Sequence,
     TypeVar,
     Union,
+    cast,
     overload,
 )
 from typing_extensions import TypeAlias
@@ -833,6 +834,38 @@ def is_undefined(obj: Any) -> TypeIs[UndefinedType]:
     return obj is Undefined
 
 
+@overload
+def _shallow_copy(obj: _CopyImpl) -> _CopyImpl: ...
+@overload
+def _shallow_copy(obj: Any) -> Any: ...
+def _shallow_copy(obj: _CopyImpl | Any) -> _CopyImpl | Any:
+    if isinstance(obj, SchemaBase):
+        return obj.copy(deep=False)
+    elif isinstance(obj, (list, dict)):
+        return obj.copy()
+    else:
+        return obj
+
+
+@overload
+def _deep_copy(obj: _CopyImpl, by_ref: set[str]) -> _CopyImpl: ...
+@overload
+def _deep_copy(obj: Any, by_ref: set[str]) -> Any: ...
+def _deep_copy(obj: _CopyImpl | Any, by_ref: set[str]) -> _CopyImpl | Any:
+    copy = partial(_deep_copy, by_ref=by_ref)
+    if isinstance(obj, SchemaBase):
+        args = (copy(arg) for arg in obj._args)
+        kwds = {k: (copy(v) if k not in by_ref else v) for k, v in obj._kwds.items()}
+        with debug_mode(False):
+            return obj.__class__(*args, **kwds)
+    elif isinstance(obj, list):
+        return [copy(v) for v in obj]
+    elif isinstance(obj, dict):
+        return {k: (copy(v) if k not in by_ref else v) for k, v in obj.items()}
+    else:
+        return obj
+
+
 class SchemaBase:
     """
     Base class for schema wrappers.
@@ -870,7 +903,7 @@ def __init__(self, *args: Any, **kwds: Any) -> None:
         if DEBUG_MODE and self._class_is_valid_at_instantiation:
             self.to_dict(validate=True)
 
-    def copy(  # noqa: C901
+    def copy(
         self, deep: bool | Iterable[Any] = True, ignore: list[str] | None = None
     ) -> Self:
         """
@@ -887,53 +920,11 @@ def copy(  # noqa: C901
             A list of keys for which the contents should not be copied, but
             only stored by reference.
         """
-
-        def _shallow_copy(obj):
-            if isinstance(obj, SchemaBase):
-                return obj.copy(deep=False)
-            elif isinstance(obj, list):
-                return obj[:]
-            elif isinstance(obj, dict):
-                return obj.copy()
-            else:
-                return obj
-
-        def _deep_copy(obj, ignore: list[str] | None = None):
-            if ignore is None:
-                ignore = []
-            if isinstance(obj, SchemaBase):
-                args = tuple(_deep_copy(arg) for arg in obj._args)
-                kwds = {
-                    k: (_deep_copy(v, ignore=ignore) if k not in ignore else v)
-                    for k, v in obj._kwds.items()
-                }
-                with debug_mode(False):
-                    return obj.__class__(*args, **kwds)
-            elif isinstance(obj, list):
-                return [_deep_copy(v, ignore=ignore) for v in obj]
-            elif isinstance(obj, dict):
-                return {
-                    k: (_deep_copy(v, ignore=ignore) if k not in ignore else v)
-                    for k, v in obj.items()
-                }
-            else:
-                return obj
-
-        try:
-            deep = list(deep)  # type: ignore[arg-type]
-        except TypeError:
-            deep_is_list = False
-        else:
-            deep_is_list = True
-
-        if deep and not deep_is_list:
-            return _deep_copy(self, ignore=ignore)
-
+        if deep is True:
+            return cast("Self", _deep_copy(self, set(ignore) if ignore else set()))
         with debug_mode(False):
             copy = self.__class__(*self._args, **self._kwds)
-        if deep_is_list:
-            # Assert statement is for the benefit of Mypy
-            assert isinstance(deep, list)
+        if _is_iterable(deep):
             for attr in deep:
                 copy[attr] = _shallow_copy(copy._get(attr))
         return copy
@@ -1240,6 +1231,13 @@ def __dir__(self) -> list[str]:
 
 TSchemaBase = TypeVar("TSchemaBase", bound=SchemaBase)
 
+_CopyImpl = TypeVar("_CopyImpl", SchemaBase, Dict[Any, Any], List[Any])
+"""
+Types which have an implementation in ``SchemaBase.copy()``.
+
+All other types are returned **by reference**.
+"""
+
 
 def _is_dict(obj: Any | dict[Any, Any]) -> TypeIs[dict[Any, Any]]:
     return isinstance(obj, dict)
diff --git a/tools/schemapi/schemapi.py b/tools/schemapi/schemapi.py
index 1c756c2a2..a77afd4a4 100644
--- a/tools/schemapi/schemapi.py
+++ b/tools/schemapi/schemapi.py
@@ -23,6 +23,7 @@
     Sequence,
     TypeVar,
     Union,
+    cast,
     overload,
 )
 from typing_extensions import TypeAlias
@@ -831,6 +832,38 @@ def is_undefined(obj: Any) -> TypeIs[UndefinedType]:
     return obj is Undefined
 
 
+@overload
+def _shallow_copy(obj: _CopyImpl) -> _CopyImpl: ...
+@overload
+def _shallow_copy(obj: Any) -> Any: ...
+def _shallow_copy(obj: _CopyImpl | Any) -> _CopyImpl | Any:
+    if isinstance(obj, SchemaBase):
+        return obj.copy(deep=False)
+    elif isinstance(obj, (list, dict)):
+        return obj.copy()
+    else:
+        return obj
+
+
+@overload
+def _deep_copy(obj: _CopyImpl, by_ref: set[str]) -> _CopyImpl: ...
+@overload
+def _deep_copy(obj: Any, by_ref: set[str]) -> Any: ...
+def _deep_copy(obj: _CopyImpl | Any, by_ref: set[str]) -> _CopyImpl | Any:
+    copy = partial(_deep_copy, by_ref=by_ref)
+    if isinstance(obj, SchemaBase):
+        args = (copy(arg) for arg in obj._args)
+        kwds = {k: (copy(v) if k not in by_ref else v) for k, v in obj._kwds.items()}
+        with debug_mode(False):
+            return obj.__class__(*args, **kwds)
+    elif isinstance(obj, list):
+        return [copy(v) for v in obj]
+    elif isinstance(obj, dict):
+        return {k: (copy(v) if k not in by_ref else v) for k, v in obj.items()}
+    else:
+        return obj
+
+
 class SchemaBase:
     """
     Base class for schema wrappers.
@@ -868,7 +901,7 @@ def __init__(self, *args: Any, **kwds: Any) -> None:
         if DEBUG_MODE and self._class_is_valid_at_instantiation:
             self.to_dict(validate=True)
 
-    def copy(  # noqa: C901
+    def copy(
         self, deep: bool | Iterable[Any] = True, ignore: list[str] | None = None
     ) -> Self:
         """
@@ -885,53 +918,11 @@ def copy(  # noqa: C901
             A list of keys for which the contents should not be copied, but
             only stored by reference.
         """
-
-        def _shallow_copy(obj):
-            if isinstance(obj, SchemaBase):
-                return obj.copy(deep=False)
-            elif isinstance(obj, list):
-                return obj[:]
-            elif isinstance(obj, dict):
-                return obj.copy()
-            else:
-                return obj
-
-        def _deep_copy(obj, ignore: list[str] | None = None):
-            if ignore is None:
-                ignore = []
-            if isinstance(obj, SchemaBase):
-                args = tuple(_deep_copy(arg) for arg in obj._args)
-                kwds = {
-                    k: (_deep_copy(v, ignore=ignore) if k not in ignore else v)
-                    for k, v in obj._kwds.items()
-                }
-                with debug_mode(False):
-                    return obj.__class__(*args, **kwds)
-            elif isinstance(obj, list):
-                return [_deep_copy(v, ignore=ignore) for v in obj]
-            elif isinstance(obj, dict):
-                return {
-                    k: (_deep_copy(v, ignore=ignore) if k not in ignore else v)
-                    for k, v in obj.items()
-                }
-            else:
-                return obj
-
-        try:
-            deep = list(deep)  # type: ignore[arg-type]
-        except TypeError:
-            deep_is_list = False
-        else:
-            deep_is_list = True
-
-        if deep and not deep_is_list:
-            return _deep_copy(self, ignore=ignore)
-
+        if deep is True:
+            return cast("Self", _deep_copy(self, set(ignore) if ignore else set()))
         with debug_mode(False):
             copy = self.__class__(*self._args, **self._kwds)
-        if deep_is_list:
-            # Assert statement is for the benefit of Mypy
-            assert isinstance(deep, list)
+        if _is_iterable(deep):
             for attr in deep:
                 copy[attr] = _shallow_copy(copy._get(attr))
         return copy
@@ -1238,6 +1229,13 @@ def __dir__(self) -> list[str]:
 
 TSchemaBase = TypeVar("TSchemaBase", bound=SchemaBase)
 
+_CopyImpl = TypeVar("_CopyImpl", SchemaBase, Dict[Any, Any], List[Any])
+"""
+Types which have an implementation in ``SchemaBase.copy()``.
+
+All other types are returned **by reference**.
+"""
+
 
 def _is_dict(obj: Any | dict[Any, Any]) -> TypeIs[dict[Any, Any]]:
     return isinstance(obj, dict)