diff --git a/starlette/middleware/base.py b/starlette/middleware/base.py
index 03afdd9c5..0be4973a5 100644
--- a/starlette/middleware/base.py
+++ b/starlette/middleware/base.py
@@ -1,6 +1,13 @@
 from __future__ import annotations
 
-from collections.abc import AsyncGenerator, AsyncIterable, Awaitable, Callable, Mapping, MutableMapping
+from collections.abc import (
+    AsyncGenerator,
+    AsyncIterable,
+    Awaitable,
+    Callable,
+    Mapping,
+    MutableMapping,
+)
 from typing import Any, TypeVar
 
 import anyio
@@ -29,7 +36,7 @@ def __init__(self, scope: Scope, receive: Receive):
         super().__init__(scope, receive)
         self._wrapped_rcv_disconnected = False
         self._wrapped_rcv_consumed = False
-        self._wrapped_rc_stream = self.stream()
+        self._wrapped_rcv_stream: AsyncGenerator[bytes, None] | None = None
 
     async def wrapped_receive(self) -> Message:
         # wrapped_rcv state 1: disconnected
@@ -80,7 +87,11 @@ async def wrapped_receive(self) -> Message:
         else:
             # body() was never called and stream() wasn't consumed
             try:
-                stream = self.stream()
+                stream = self._wrapped_rcv_stream
+                if stream is None:
+                    stream = self.stream()
+                    self._wrapped_rcv_stream = stream
+
                 chunk = await stream.__anext__()
                 self._wrapped_rcv_consumed = self._stream_consumed
                 return {
@@ -103,7 +114,7 @@ async def __call__(self, scope: Scope, receive: Receive, send: Send) -> None:
             await self.app(scope, receive, send)
             return
 
-        request = _CachedRequest(scope, receive)
+        request: _CachedRequest = scope.setdefault("__starlette_CachedRequest__", _CachedRequest(scope, receive))
         wrapped_receive = request.wrapped_receive
         response_sent = anyio.Event()
         app_exc: Exception | None = None
diff --git a/tests/middleware/test_base.py b/tests/middleware/test_base.py
index 1b0b94760..875b031d1 100644
--- a/tests/middleware/test_base.py
+++ b/tests/middleware/test_base.py
@@ -15,7 +15,12 @@
 from starlette.middleware import Middleware, _MiddlewareFactory
 from starlette.middleware.base import BaseHTTPMiddleware, RequestResponseEndpoint
 from starlette.requests import ClientDisconnect, Request
-from starlette.responses import FileResponse, PlainTextResponse, Response, StreamingResponse
+from starlette.responses import (
+    FileResponse,
+    PlainTextResponse,
+    Response,
+    StreamingResponse,
+)
 from starlette.routing import Route, WebSocketRoute
 from starlette.testclient import TestClient
 from starlette.types import ASGIApp, Message, Receive, Scope, Send
@@ -106,6 +111,72 @@ def test_custom_middleware(test_client_factory: TestClientFactory) -> None:
         assert text == "Hello, world!"
 
 
+@pytest.mark.anyio
+async def test_streaming_request(
+    test_client_factory: TestClientFactory,
+) -> None:
+    async def echo(request: Request) -> Response:
+        body = await request.body()
+        return Response(
+            body,
+            status_code=200,
+            media_type=request.headers.get("Content-Type", "text/plain"),
+        )
+
+    async def skip_dispatch(request: Request, call_next: RequestResponseEndpoint) -> Response:
+        return await call_next(request)
+
+    app = Starlette(
+        routes=[Route("/echo", echo)],
+        middleware=[Middleware(BaseHTTPMiddleware, dispatch=skip_dispatch)],
+    )
+
+    async def receive_generator() -> AsyncGenerator[Message, None]:
+        yield {"type": "http.request", "body": b"first chunk\n", "more_body": True}
+        yield {"type": "http.request", "body": b"second chunk", "more_body": False}
+        yield {"type": "http.disconnect"}
+
+    async def send(message: Message) -> None:
+        sent.append(message)
+
+    sent: list[Message] = []
+
+    scope = {
+        "type": "http",
+        "version": "3",
+        "method": "GET",
+        "path": "/echo",
+        "headers": [[b"Content-Type", "text/plain"]],
+    }
+
+    receive = receive_generator().__anext__
+
+    await app(scope, receive, send)
+
+    assert sent == [
+        {
+            "type": "http.response.start",
+            "status": 200,
+            "headers": [
+                (b"content-length", b"24"),
+                (b"content-type", b"text/plain; charset=utf-8"),
+            ],
+        },
+        {
+            "type": "http.response.body",
+            "more_body": True,
+            "body": b"first chunk\nsecond chunk",
+        },
+        # BaseHTTPMiddleware converts responses into StreamingResponses that yield b"" at end
+        {"type": "http.response.body", "more_body": False, "body": b""},
+    ]
+
+    assert await receive() == {"type": "http.disconnect"}
+
+    with pytest.raises(StopAsyncIteration):
+        await receive()
+
+
 def test_state_data_across_multiple_middlewares(
     test_client_factory: TestClientFactory,
 ) -> None:
@@ -298,7 +369,9 @@ async def send(message: Message) -> None:
     assert background_task_run.is_set()
 
 
-def test_run_background_tasks_raise_exceptions(test_client_factory: TestClientFactory) -> None:
+def test_run_background_tasks_raise_exceptions(
+    test_client_factory: TestClientFactory,
+) -> None:
     # test for https://github.com/Kludex/starlette/issues/2625
 
     async def sleep_and_set() -> None: