Whenever Nafs Asp deals damage to a player, that player loses 1 life at the beginning of their next draw step unless they pay {1} before that draw step.
If it damages a player twice, they get to pay or take damage twice during their next draw step.
If its damage gets redirected to its controller, it will still trigger the ability.
The ability does not cause itself to trigger again. It cause loss of life, and not damage.
The player can pay the {1} mana at any time after damage is done before the draw step of that player’s turn. People commonly pay during upkeep.