Between you and me, this person is not such a great friend. She is right in understanding that, "Between you and I," is not grammatically correct, but she is not right about much else.
Her objective in correcting you in public certainly does not emanate from her desire to hear English spoken in its purest form. Rather, her objective is probably to make you look weak and to make herself look strong.
In order to put a stop to this behavior, you need to do more than tell her that you don't appreciate it. This situation calls for some assertiveness. The next time she goes grammatical on you, try to meet with her as soon as possible, obviously in private. Tell her that you regard her public comments as insulting and hurtful, and you want them to stop right now. Let her know that it is very hard for you to remain friends with someone who treats you this way.
If her public grammar lessons continue, then perhaps it is time for you to teach her a lesson about losing a friend.
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