Who Was That Lady? (1960) on Netflix

Ann Wilson catches her straight-laced husband, Columbia University Assistant Professor of Chemistry David Wilson, kissing another woman. From David's perspective, he was the one being kissed innocently, the woman in question being a grateful transfer student. However, Ann wants a divorce. On the advice of David's friend, TV writer Mike Haney who is a self-professed libertine, David tries to convince Ann that he is really an FBI agent, the kiss all in the name of national security. David, with the help of Mike's prop friend, even gets a fake ID and gun. The ruse works. But Ann, excited about her husband's secret life, may not be able to keep her husband's occupation a secret. And the real FBI get wind that a fake ID has been made. They may be able to use David to their advantage. But if someone like Ann believes her husband is an FBI agent, others, such as foreign spies, may believe so as well.