I'm no historian (maybe I should stop here?) but in my readings I've traced the suicide element of warfare in Islam as far back as the 11th century with Hasan Bin Sabbah's Secret Order of the Hashishins. The term Hashishin is the origin of the English word 'assassin' and this order is what the original Assassin's Creed is based on (it digresses quite a bit in it's sequels). They were an off-shoot of Shia'ism and focused on political assassinations. They would secretly join armies and ranks of their targets and work their way up to get close enough to assassinate. The point was to let it be known who the responsible group was so these assassins would announce their true identity after the job was done, making every mission virtually a suicide mission (this also led to their downfall).

After this, I haven't really seen too much more evidence of suicide attacks until towards the end of the Islamic golden age c. late-1400s.

This is of course regarding Islam alone.