Is it permissible to keep a picture of a girl after her mother refused their marriage, knowing that her father approved?

The validity of a marriage is contingent upon the approval of the woman's guardian, who is her father. The mother's permission is not a prerequisite, but consulting her is recommended. If the father approves, he can solemnize the marriage.

Regarding proceeding with this marriage: If the girl is of satisfactory religious and moral character, and the mother's refusal is due to minor reasons that can be overcome, then there is no harm. However, if the refusal is due to a matter that will not soon dissipate, it is preferable to abandon the marriage due to the potential for discord it may cause. The matter requires a balance between benefits and harms.

As for keeping a photograph of a non-mahram girl, it is not permissible because she is a non-mahram, and it is not lawful to look at her. This is a gateway to temptation that contradicts the command to lower one's gaze, as stated in His Almighty's words: "Tell the believing men to reduce [some] of their vision and guard their private parts. That is purer for them. Indeed, Allah is Acquainted with what they do." (An-Nur/30).

Secondly: It is not permissible to keep photographs for remembrance, due to the general evidence that prohibits photography and sternly warns against it, except in cases of necessity and dire need.

You must dispose of the photograph and seek Allah's guidance through Istikhara before proceeding with the marriage.