Do the oppenents have to assign one damage to one unit in the battlefield or one damage to every unit in the battlefield?

1 damage to any unit, not every unit.

Ok thanks

But I think It's a valid question.

My native language isnt english so I looked it up

And if you use any as an pronomen it means every or each.

Now I have found another post about this

http://community.fantasyflightgames.com/index.php?/topic/28281-shadow-warrior-question/?hl=%20shadow%20%20warrior