diff --git a/Lib/test/test_import/__init__.py b/Lib/test/test_import/__init__.py
index 14a688de1c6..1e33274b873 100644
--- a/Lib/test/test_import/__init__.py
+++ b/Lib/test/test_import/__init__.py
@@ -278,6 +278,7 @@ class ImportTests(unittest.TestCase):
             """))
         script_helper.assert_python_ok(testfn)
 
+    @skip_if_dont_write_bytecode
     def test_timestamp_overflow(self):
         # A modification timestamp larger than 2**32 should not be a problem
         # when importing a module (issue #11235).
diff --git a/Misc/NEWS b/Misc/NEWS
index 4299e4e1db2..78980fe9663 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -319,6 +319,9 @@ Tests
 - Issue #25616: Tests for OrderedDict are extracted from test_collections
   into separate file test_ordered_dict.
 
+- Issue #26583: Skip test_timestamp_overflow in test_import if bytecode
+  files cannot be written.
+
 Build
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