diff --git a/equinox/nn/_stateful.py b/equinox/nn/_stateful.py
index 5a992d39..fc9b9ab0 100644
--- a/equinox/nn/_stateful.py
+++ b/equinox/nn/_stateful.py
@@ -1,3 +1,4 @@
+import types
 from collections.abc import Callable
 from typing import Any, Generic, TYPE_CHECKING, TypeVar, Union
 from typing_extensions import ParamSpec
@@ -52,6 +53,15 @@ def __init__(self, init: _Value):
 
         - `init`: The initial value for the state.
         """
+        if isinstance(init, types.FunctionType):
+            # Technically a function is valid here, since we could allow any pytree.
+            # In practice that's weird / kind of useless, so better to explicitly raise
+            # the deprecation error.
+            raise ValueError(
+                "As of Equinox v0.11.0, `eqx.nn.StateIndex` now accepts the value "
+                "of the initial state directly. (Not a function that creates the "
+                "initial state.)"
+            )
         self.marker = object()
         self.init = init