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It’s heartbreaking to watch our daughters be kind to others, only to be treated poorly or rejected in return. Making new friends doesn’t come easily to every child. I thanked her for being honest with me and sharing what was going on; we’ve talked quite a bit. I explained to her that a true friend would never suggest that she harm herself. I don’t want to be too pushy by forbidding the friendship or coming across as overly negative; instead, I need to gradually distance them—in a way that my daughter won’t notice. If I were to say what I’m really thinking, I’d likely destroy the trust she has in me; she might even think I’m being cruel or something similar—especially since this is the first friend she’s made since we returned to Venezuela 18 months ago.
In terms of the general question. My daughter had a couple friends in her dance class who I thought were not treating her nicely. My daughter was being very friendly and kind to them but they were very dismissive with her because she is not part of their friend group from school (my daughter is homeschooled). I just told my daughter "you can be friends with whoever you want but you should not go out of your way to try to please and be friends with people who don't treat you nicely."
In your situation I would definitely talk to the other girl's parents and let them know what was going on. Suggesting she hurt herself and try to blame it on her father as retribution for being upset with him is crossing the line.
Personally I would suggest you tell your daughter it was smart and mature of her to tell you and that she can be friends with whoever she wants but she doesn't need to be friends with people who are looking to involve her in mischief and that you trust her to make good decisions about who to be friends with.
See where it goes from there.