It would seem to me that suicide in a tribal/community-based species is the one of the most counter-productive actions a member could take, considering the interdependence that exists within societies of that kind. Also, I'd assume that conceiving and following through with suicidal thoughts requires a degree of autonomy and self-awareness that is uncommon in the animal kingdom. I want to say that I've seen articles about suicidal behavior seen in dogs and dolphins, but I feel that those cases could either be just anecdotal/inaccurate, or they occurred only because the specific animals in question were not a part of packs or groups that interdependence would have existed in (domesticated dogs obviously don't belong to any such group, and iirc the dolphin was in captivity without a pod).

Is there a simpler answer to this that I'm not seeing? If there was a distinct turning point in humanity's ability to commit suicide, when was it? Or has suicidal behavior been observed observed for as long as we can tell? Is there even a way to find this out?