There appears to be a bundle of evidence pointing to the fact that psychiatric drugs shorten life-span generally. Also and more specifically, neuroleptic malignant syndrome caused by antipsychotics, serotonin syndrome caused by antidepressants, etc., can be fatal. I'm guessing then that the deaths caused by mental illness can be attributed to the medications mentally ill people sometimes take. Side effects, if you will.
https://www.psychologytoday.com/ca/blog/science-isnt-golden/201109/full-disclosure-needed-about-psychiatric-drugs-shorten-life