diff --git a/pkgs/development/python-modules/websockets/default.nix b/pkgs/development/python-modules/websockets/default.nix
index 907024735afb..918cb54ae5a6 100644
--- a/pkgs/development/python-modules/websockets/default.nix
+++ b/pkgs/development/python-modules/websockets/default.nix
@@ -24,26 +24,31 @@ buildPythonPackage rec {
 
   nativeBuildInputs = [ setuptools ];
 
-  patchPhase = ''
-    # Disable all tests that need to terminate within a predetermined amount of
-    # time. This is nondeterministic.
-    sed -i 's/with self.assertCompletesWithin.*:/if True:/' \
-      tests/legacy/test_protocol.py
+  patchPhase =
+    ''
+      # Disable all tests that need to terminate within a predetermined amount of
+      # time. This is nondeterministic.
+      sed -i 's/with self.assertCompletesWithin.*:/if True:/' \
+        tests/legacy/test_protocol.py
 
-    # Disables tests relying on tight timeouts to avoid failures like:
-    #   File "/build/source/tests/legacy/test_protocol.py", line 1270, in test_keepalive_ping_with_no_ping_timeout
-    #     ping_1_again, ping_2 = tuple(self.protocol.pings)
-    #   ValueError: too many values to unpack (expected 2)
-    for t in \
-             test_keepalive_ping_stops_when_connection_closing \
-             test_keepalive_ping_does_not_crash_when_connection_lost \
-             test_keepalive_ping \
-             test_keepalive_ping_not_acknowledged_closes_connection \
-             test_keepalive_ping_with_no_ping_timeout \
-      ; do
-      sed -i "s/def $t(/def skip_$t(/" tests/legacy/test_protocol.py
-    done
-  '';
+      # Disables tests relying on tight timeouts to avoid failures like:
+      #   File "/build/source/tests/legacy/test_protocol.py", line 1270, in test_keepalive_ping_with_no_ping_timeout
+      #     ping_1_again, ping_2 = tuple(self.protocol.pings)
+      #   ValueError: too many values to unpack (expected 2)
+      for t in \
+               test_keepalive_ping_stops_when_connection_closing \
+               test_keepalive_ping_does_not_crash_when_connection_lost \
+               test_keepalive_ping \
+               test_keepalive_ping_not_acknowledged_closes_connection \
+               test_keepalive_ping_with_no_ping_timeout \
+        ; do
+        sed -i "s/def $t(/def skip_$t(/" tests/legacy/test_protocol.py
+      done
+    ''
+    + lib.optionalString (pythonOlder "3.11") ''
+      # Our Python 3.10 and older raise SSLError instead of SSLCertVerificationError
+      sed -i "s/def test_reject_invalid_server_certificate(/def skip_test_reject_invalid_server_certificate(/" tests/sync/test_client.py
+    '';
 
   nativeCheckInputs = [ unittestCheckHook ];