Is it permissible for a married woman who has fallen in love with a foreign man and committed an unlawful act with him to ask her husband for a divorce in order to marry the other man, and is it permissible for any relationship to remain between them if she does not divorce, or even after her divorce?

You must immediately initiate a sincere repentance, for the crime of zina (fornication/adultery) is a major sin. It becomes even more sinful if the woman involved is married, due to the betrayal of her husband. The Prophet, peace and blessings be upon him, forbade inciting a woman against her husband. Some scholars even went so far as to prohibit the marriage of the inciter to the woman he incited, and even ruled that his marriage to her is invalid. Repentance has conditions, and one must increase in acts of obedience so that Allah may accept the repentance. It is not permissible for a woman to seek divorce without a legitimate Shar'i excuse, and it is forbidden to establish an emotional relationship with a non-mahram woman, whether she is married or divorced. The relationship with her must be completely severed.