The only person who can answer your question is the girl's mother. My granddaughter has a peanut allergy and we are careful that she doesn't eat anything with peanuts in it. She reads labels and is able to say, "I can't eat that." That is the only way the allergy affects her life. Her mother and brother both eat peanut butter. She has never had a negative reaction to peanuts.
Her peanut allergy is relatively minor. There are serious peanut allergies in which the person can't even be in the same room with peanuts. You need to know how serious her allergy is and what you should do. Since she didn't tell you specifically what to do I'd guess that you just need to not feed her anything with peanuts but you need to check with her mother.
Please relax and enjoy your daughter enjoying her. I understand your concern. There has been so much publicity about peanut allergies. For some people the allergy can be life threatening but the majority of people with allergies are not that sensitive, in part because they don't expose themselves to peanut butter. It is best to be careful but it's counter productive to let it scare people so much that the child is treated differently when it's not necessary.
I have a peanut allergy but I'm not as careful as my granddaughter and her mother. Every once in awhile I eat something with peanuts in it and my throat and ear canals gets itchy. Often that is the only way I knew that the product contained peanuts.
I'm also allergic to chocolate and most fruits and veggies when they're raw. Cooked seems to be OK. So I try to stay away from foods that I'm allergic to but I don't worry about it. It's highly unlikely that I'd have a major reaction.
I was diagnosed before law suits took over. A doctor probably could not tell a patient that a peanut allergy was unlikely to be life threatening today. I think it is true that allergies to peanuts and shellfish are more likely to be life threatening than most other allergies.
Guess I got on my soap box and gave you more info than you need. I did so because I would like people to know that although food allergies should be taken seriously they don't always have to be feared.