Let her break up with you, honestly.

I was in your position just a little over a week ago, my boyfriend (now ex) was in your girlfriend's position. He didn't have depression (I think?) but he had a shit ton of issues going on in his personal life, and he told me that he felt that he couldn't give me everything I deserved at the moment, because his mind was too focused on his own personal issues; after we got into a disagreement about how he snaps at me whenever he was in a bad mood (with things unrelated to me). I was extremely happy with him, but he wanted to break up with me. It's really hard to take a break up, when you don't want it to happen, not gonna lie.

I wouldn't give up on your girlfriend, no. Honestly, if she really wants to break up, agree to it. Make it clear that you'll be there for her as a close friend, and that you're willing to talk to her through any issues she's having. Check up on her occasionally to see how she's doing, that kind of thing. That's what I've been trying to do.