My friend is obsessed with my mum!?

So I have a friend she’s kinda really weird I became friends with her because I noticed she had none and felt bad but she has a lot of mental issues and stuff. Anyway last week I invited her around to my house for drinks we had a few glasses of wine and my mum asked if she had a boyfriend and she was explaining that she didn’t know if she liked girls or boys my mum told her it’s fine to experiment and was telling her about she was younger that she used to experiment with both. My friend began to ask creepy questions like would she want to be with a girl again and would she ever have sex with another girl and stuff like that. Anyway she was supposed to be going home in a taxi and I told her it was time to go she started to pretend she had no money even though I knew she did because I went to the cash machine earlier and she was like cant I just stay here I feel comfortable here and was really dragging it out to stay so my mum let her stay. Since then she has not stopped talking about my mum and I was In work last night on my night shift and my mum rang me saying that my friend was at my house claiming I’d invited her over for a sleep over!? I clearly had not done this and when my mum tried to send her Away she was begging to sleep saying she had come all this way for nothing. She’s obsessed what can I do??