Not in the original Super Mario Bros. You run across the bridge, dodge Bowser's attacks, and "win" by reaching the Axe on the far side. The strong implication (within the NES's limited visuals) is that by collecting the axe, you have destroyed the bridge and sent Bowser plummeting into the lava. Alternately, you can hit him repeatedly with Fireballs, and he will eventually die like any other enemy.

Mario absolutely "kills" Bowser in the original games. Though, of course you're quite correct that he returns in the sequels, and it's heavily implied that the Koopa clan does not die upon contact with lava, unlike the rest of the creatures in the kingdom.

https://www.mariowiki.com/Axe