What is the ruling of Islamic law regarding a divorced woman who engaged in a romantic relationship with a non-mahram man and then fled from her father's house after he discovered the relationship and intended to kill her, and what are the consequences of this, whether she is with this man or in another place?

Killing is a grave crime, especially the premeditated killing of a believer, for which the punishment is Hellfire. A believer remains at ease concerning his religion as long as he has not committed unlawful killing. A father threatening to kill his daughter is a serious matter that is not permissible for him, even if she committed adultery while being muhsanah (married or previously married and having had intercourse), because the implementation of hudud (punishments prescribed by Islamic law) is the responsibility of the ruler and not individuals. The father must be reminded and warned to fear Allah, and the daughter should be advised to repent and abandon the forbidden relationship. If the father insists on killing, the matter must be referred to the Sharia judge.