TS, it sounds like this girl might have more problems than just depression. Her behaviour seems to be a bit beyond that which is symptomatic of typical depression.
Anyway, to answer the question, yes, I have dated someone with depression, namely my wife. We’ve been together since 2006 and she was diagnosed with depression 2011 and was on antidepressants until 2016. In hindsight she showed the symptoms of depression in 2009, I was just too stupid to really notice them or to much of a pussy to discuss my concerns with her then.
My advice would be to stay away if you’ve only been with this girl a few months as being with someone who’s depressed is frustrating, exhausting and a lot of work. I stayed with my wife because I love her and am firmly committed to her, but if you’re not that invested in the relationship yet (ie not married, or have kids or whatever) then you really need to think hard on this. While leaving someone with depression seems like a selfish, assholish move, people with depression typically bring those around them down with them. You need to be aware of that, and strong enough to support yourself before you’re able to effectively support the other person who’s suffering from depression.
Unless you’re fully prepared to handle being with someone with depression you’re doing both of you a disservice by trying to salvage a relationship that’s doomed to failure and which could leave your own life in ruin.