Is it permissible to keep a picture in which a female friend appears with a male accompanying her (her cousin or a strange man), knowing that the intention is to keep the picture because the friend is in it, and that Allah Almighty said: "And tell the believing women to lower their gaze..."?

It is safer not to acquire a picture that includes a non-mahram (definition) man, as it is an entry point for Satan and a door to temptation. Satan lies in wait for humans, and Allah the Almighty has warned against following his footsteps. If there is a need for a picture of your female friend, a separate picture of her can be requested, bearing in mind that there is a difference of opinion among scholars regarding photography and acquiring photographs.