I'm thinking of cutting my friendship to a girl who I think betrayed my trust
What did she do? Are you sure she did that in the past?
She told a lot of lies about me and then she downplayed it saying that it's not a big deal.
It's fine to stand up for yourself when people are mean to you. If you don't she may treat you poorly forever.
I get it, but at the same time, how should I approach this? Should I confront her?
Just cut it off completely. You don't even have to say anything if you wish it to go that way.