It's something of a question of terminology. So suicidal attacks, in which the attacker does not expect to survive but dying isn't actually a part of the goal is in all likelihood pre-historic. There are also some classical examples of suicide attacks in which ththe attackers do seek death as part of their attack.

Bombings are a newer phenomenon. There are early bomb attacks in which the attacker accidentally blows themselves up. And again suicidal attacks where the attacker does not reasonably seek to survive but dying isnt part of the goal per se.

The first suicide bombings in the proper sense, where death is part of the attack, for instance by strapping the bombs to your own body, arise in the late 19th early 20th century with the Russian anarchist bombers.

Islamic suicide bombing is a newer phenonomenon still, and was first proposed by the Muslim Brotherhood in the 1950s in a bid to assassinate Nasser. This was rejected as too extreme and they instead opted to try to shoot him.

I've never been able to definitively confirm when the first Islamist suicide bombing took place, but the earliest I've conclusively found are proto-Hezbollah's attacks in Lebanon in the early 80s, though I suspect there are earlier examples I just haven't found. They don't really take off though until Palestinian groups start using them in the early 90s (first attack was 1989.)

Edit: proto, not pronto