Direct answer: The B. Free Franklin Post Office in Philadelphia is widely cited as the unique U.S. post office that does not fly the American flag. Context and notes:

• The B. Free Franklin Post Office is housed in a historic Philadelphia building associated with Benjamin Franklin. It was established to commemorate Franklin’s era and the colonial postal service, and it intentionally does not display the U.S. flag or use a ZIP code, aligning with its thematic dedication to colonial times. This makes it the only post office in the United States with that combination of features.

If you’d like, I can pull additional sources or provide a concise summary of what makes this office unique (history, signage, dress code, and visitor access) with citations.