Card abilities must finish before other cards can be played, yes, but just like any other card causing you to lose the battle, you get special initiative to interrupt/negate it.
It's the same principle as being discarded, for example. By the time the discard resolves, your character would be in the discard pile where enhancements couldn't be played on them, but you get a chance to interrupt or negate the discard before you're actually discarded.
