Is it permissible to continue a romantic relationship with the aim of marriage with a girl whose family refuses due to ethnic differences, knowing that both parties desire to marry?

Islam prohibits the establishment of a relationship between a man and a woman who is a stranger to him, as this invites temptation and corruption.

If you wish to marry her, you may try to convince her family to agree.

However, if her family, especially her parents, refuse the marriage, then she must obey her parents, and both of you should separate from each other. For whoever gives up something for the sake of Allah, Allah will compensate him with something better.