diff --git a/src/anthropic/lib/_retry.py b/src/anthropic/lib/_retry.py
index bf66c744..3eaea0be 100644
--- a/src/anthropic/lib/_retry.py
+++ b/src/anthropic/lib/_retry.py
@@ -40,9 +40,22 @@ def jitter(low: float, high: float) -> float:
 
 
 def is_fatal_status_error(err: Exception) -> bool:
-    """True for a 4xx that retrying will not fix (bad key, missing resource).
+    """True for a status error that retrying will not fix (bad key, missing resource).
 
-    Aligns with the core client's ``_should_retry`` policy: 408 / 409 / 429 are
-    transient and worth retrying; every other 4xx is fatal.
+    Aligns with the core client's ``_should_retry`` policy:
+
+    * the server's explicit ``x-should-retry`` header wins — ``true`` is never
+      fatal, ``false`` is always fatal, regardless of the status code;
+    * otherwise 408 / 409 / 429 are transient and worth retrying, and every
+      other 4xx is fatal.
     """
-    return isinstance(err, APIStatusError) and 400 <= err.status_code < 500 and err.status_code not in _RETRYABLE_4XX
+    if not isinstance(err, APIStatusError):
+        return False
+
+    should_retry_header = err.response.headers.get("x-should-retry")
+    if should_retry_header == "true":
+        return False
+    if should_retry_header == "false":
+        return True
+
+    return 400 <= err.status_code < 500 and err.status_code not in _RETRYABLE_4XX
diff --git a/tests/lib/test_retry.py b/tests/lib/test_retry.py
new file mode 100644
index 00000000..d2d81033
--- /dev/null
+++ b/tests/lib/test_retry.py
@@ -0,0 +1,48 @@
+from __future__ import annotations
+
+import httpx
+import pytest
+
+from anthropic.lib._retry import is_fatal_status_error
+from anthropic._exceptions import APIError, APIStatusError, APIConnectionError
+
+
+def _status_error(status_code: int, headers: dict[str, str] | None = None) -> APIStatusError:
+    response = httpx.Response(
+        status_code,
+        headers=headers or {},
+        request=httpx.Request("POST", "https://api.anthropic.com/v1/messages"),
+    )
+    return APIStatusError("boom", response=response, body=None)
+
+
+class TestIsFatalStatusError:
+    @pytest.mark.parametrize("status_code", [400, 401, 403, 404, 422])
+    def test_plain_4xx_is_fatal(self, status_code: int) -> None:
+        assert is_fatal_status_error(_status_error(status_code)) is True
+
+    @pytest.mark.parametrize("status_code", [408, 409, 429])
+    def test_transient_4xx_is_not_fatal(self, status_code: int) -> None:
+        assert is_fatal_status_error(_status_error(status_code)) is False
+
+    @pytest.mark.parametrize("status_code", [500, 502, 503])
+    def test_5xx_is_not_fatal(self, status_code: int) -> None:
+        assert is_fatal_status_error(_status_error(status_code)) is False
+
+    def test_x_should_retry_true_overrides_fatal_4xx(self) -> None:
+        # the server explicitly asks to retry a status that would otherwise be fatal
+        assert is_fatal_status_error(_status_error(400, {"x-should-retry": "true"})) is False
+
+    def test_x_should_retry_false_overrides_transient_4xx(self) -> None:
+        # the server explicitly asks NOT to retry a status that would otherwise be transient
+        assert is_fatal_status_error(_status_error(429, {"x-should-retry": "false"})) is True
+
+    def test_x_should_retry_false_makes_5xx_fatal(self) -> None:
+        assert is_fatal_status_error(_status_error(503, {"x-should-retry": "false"})) is True
+
+    def test_non_status_errors_are_not_fatal(self) -> None:
+        # transport-level errors are retryable, not fatal
+        request = httpx.Request("POST", "https://api.anthropic.com/v1/messages")
+        assert is_fatal_status_error(APIConnectionError(request=request)) is False
+        assert is_fatal_status_error(APIError("x", request=request, body=None)) is False
+        assert is_fatal_status_error(ValueError("not an api error")) is False