It's hard to provide a detailed answer without seeing your code. That said, you probably just need a simple boolean variable to indicate when the enemy has been killed. At the point where you know the enemy has been shot, set the variable. Then, in the enemy's shooting code, only shoot if that variable isn't set...
Having no idea what your code looks like, here's a mock-up...
var enemy_dead = false # enemy not dead by default
if area.is_in_group("bullet"): # enemy hit by bullet
enemy_dead = true # set the enemy_dead boolean
$EnemyAnimation.play("death") # play the enemy death animation
if !enemy_dead: # if the enemy hasn't been flagged as "dead", shoot