I don't think frost bite and infections are quite the same. An infection will spread through her bloodstream towards her heart (I speak from experience), as opposed to frost bite with only kills the frozen tissue.
I hope that I am wrong, but I don't think this can heal itself. :hit
I agree with @BantyChooks, but if your nervous about having to do it with your bare hands, you could always use a gun. I am so sorry that we have to discuss this😥
I can't think of any antibiotics off the top of my head, maybe someone else will chime in and help.
Personally, I think amputation sounds inhumane. Unless you could put her an an anesthetic, it would hurt her a lot to cut it off. She would also be at risk for excessive blood loss, and she could...