Is it permissible for the wife to swear that she did not speak with a person who tried to approach her, knowing that this person is threatening to expose her with fake call recordings and is spreading malicious rumors about her to her husband and acquaintances?

A woman should cut off ties with a non-mahram man from the outset. It is not permissible for a woman to speak with a non-mahram man except with her husband's permission. If such an interaction occurs, she must sever the relationship and change her contact information. If there is no way out except to swear an oath not to speak with him, then there is no harm in her swearing the oath while using ta'areedh (double entendre/ambiguity), such as intending that she did not speak to him at that specific moment. However, this is only permissible in cases of extreme necessity. In this situation, the oath may not be effective, and it is better to be frank with the husband about some of what happened, while informing him that the relationship was severed when the man overstepped the bounds of decency.