For urgent first aid, please use the Berlin Crisis Center (Krisendienst): https://www.berliner-krisendienst.de/en/

Or if you are thinking of harming yourself, please find 24 hour help here.

You may also find a helpful service here: https://www.berlin.de/willkommenszentrum/en/everyday-life/help-and-support/

All three Berlin Universities offer counseling and psychological services. Students may contact them to discuss a range of issues from study motivation, orientation at university, meeting deadlines and exam nerves, to study overload, homesickness, family problems, depression or any other academic or non-academic issue. The current COVID-19 pandemic has notoriously caused negative effects on students’ mental health, thus making them less motivated to study and increasing symptoms of depression and isolation.
At the first meeting, the counselor will outline what steps can be taken to solve a specific problem. Further appointments can be made or else the counselor will help the student to find a therapist if necessary. The counseling sessions are strictly confidential.
https://www.fu-berlin.de/en/sites/studienberatung/psychologische_beratung/
https://www.hu-berlin.de/en/studies/counselling/psyber
https://www.studienberatung.tu-berlin.de/menu/beratung/psychologische_beratung/parameter/en/

If you are seeking long-term/ regular psychotherapy:

If you have public German health insurance, you can make an appointment with any therapist that is in the public health insurance network: https://www.kvberlin.de/fuer-patienten/arzt-und-psychotherapeutensuchePublic health insurance providers cover initial "getting to know each other" consultations (usually up to five per therapist) without any diagnosis/ extra application. If you feel comfortable with the therapist and they can take you on as a regular patient, the therapist will send an application to your health insurance provider. For this application, you will need to get a diagnosis from a general practitioner ("Hausarzt" or "Allgemeinmediziner").
However, it can be quite difficult to find a therapist who is not at full capacity and covered by the public health insurance providers. You should always double check with your therapist whether they are covered by public health insurance, what kind of reimbursements are possible or whether you have to pay privately.

If you have private health insurance or are using the EHIC from your home country, please check in with your insurance first to find out what they will cover and how to go about it. You most likely will need to contact a general practitioner ("Hausarzt" or "Allgemeinmediziner") first to get a diagnosis and then find a therapist afterwards.
It may also help to contact your health insurance provider to ask for advice on finding the right doctor or therapist and how to get the therapy covered by your insurance.

Two BMS students took the time to collect all information on how to find help for your mental health in this pdf (2021): Guide

Here are some more resources on mental health:
https://dragonflymentalhealth.org/
https://www.scholar-minds.net
https://www.studentminds.org.uk/
https://www.zeit.de/gesundheit/2021-07/psychotherapie-psychische-erkrankungen-therapieplatz-finden-therapiearten-kostenuebernahme-tipps/
https://psych-info.de/

You can also find some apps that might help you get a therapist or offer courses, however, these are mostly in German and not for free:
https://www.selfapy.com
https://de.deprexis.com

Information in German: https://www.psychotherapeutenkammer-berlin.de/ratsuchende/psychotherapie