diff --git a/panel/chat/interface.py b/panel/chat/interface.py
index 11f3e431a9..0acbe44188 100644
--- a/panel/chat/interface.py
+++ b/panel/chat/interface.py
@@ -740,5 +740,5 @@ def stream(
             # so only set to the default when not a ChatMessage
             user = user or self.user
             avatar = avatar or self.avatar
-        message_params["show_edit_icon"] = message_params.get("show_edit_icon", user == self.user)
+        message_params["show_edit_icon"] = message_params.get("show_edit_icon", user == self.user and message_params.get("edit_callback"))
         return super().stream(value, user=user, avatar=avatar, message=message, replace=replace, **message_params)