((((((((((ganymede))))))))))
Are you on medication or have you seen a doctor about your depression? If you are on medication, showing your prescription to your professor or getting a doctor's note should serve as enough "proof" of a medical need for extension on your work. If not, is your student support office able to write a letter to your professors/department explaining that you're having personal problems that may have a negative impact on your academic performance? I got a letter like that sent to my department last year because my school has a very strict attendance policy and I was unable to get to most of my classes while the depression was really bad.
It's REALLY hard to be depressed and a student at the same time. I understand what you're going through. No matter how much you want to, you can't seem to motivate yourself to get through your work. I get the sense that I'm floundering all the time, and I'm on the upswing from my depression. When I was really low, I was barely functioning at all. Depression is a complicated illness and it's not understood by the vast majority of people. Unfortunately there's a stigma and lack of education surrounding it that can just make things worse for its sufferers. I know it's easier said than done, but try not to be ashamed of your depression. You have a medical condition you can't control, any more than you'd be able to control being diabetic. Something is wrong, and you need help, and there's nothing wrong with that whatsoever. If you haven't seen a doctor, please consider going to one ASAP. It will really help to have a professional on your side right now.
As far as I'm aware, you don't need to tell anyone you're depressed if you don't want to. My individual tutors and professors are aware that I have a medical condition, but it goes no further than that, and I don't believe they're permitted to ask for specifics. (I could be wrong about that, though, so you might want to check that with your student support office.) Do you go for regular counselling sessions or would you consider doing that? I set up weekly sessions with my counsellor, both to work on my depression and learn new coping skills to help get me through the school year. Maybe you and your counsellor could develop an action plan to help see you through the rest of the semester? If there's no one in your student support office you feel you can talk to, you can also ask your doctor for a referral to a therapist.
The other thing that worked for me was trying to get a handle on things in advance of a complete breakdown. The earlier you give your professors notice of a need for an extension, the more likely they'll be to accommodate you. For instance if I realize I have three assignments due in one week, I will see about getting them spaced out a bit to keep my stress levels down and give me time to really concentrate on the individual assignments themselves. It might be too late now but maybe it's something you can consider in future.

Good luck with everything. I know how hard it is. Please believe me when I tell you that this won't last forever and you're not alone.