If a person in your life pushes you, and you've offered to compromise or discuss, then they want to win, and they want you to lose.

You are no longer in a friendly discussion.  They are preferring a hostile resolution.

Tell them no.  Once they take a hostile position, you must not give in - and if you fear that they will become upset, you must accept this possibility.  Because, once a person wants to win and wants you to lose, the worst thing you can do is give in.