Joi Lansing and Barbara Nichols in Who Was That Lady (1960)

In order to get back into the good graces with his wife with whom he has had a misunderstanding, a young chemistry professor concocts a wild story that he is an undercover FBI agent. To help him with his story he enlists the aid of a friend who is a TV writer. The wife swallows the story and later real FBI agents and enemy spies become involved in the scheme.