After quick bit of research: "Yes in fact in the original
Batman comics in 1939 he
did kill and he
killed pretty ruthlessly and he even used a gun. Later on in several
comics he is seen
killing"because he has no choice" or occasionally because he thought the person was too dangerous to let live."
Batman 1989, throws one of Joker's henchmen off the bell tower, killing him. He dropped Joker off the tower, killing him.
Batman Returns 1992, kills a thug with his own dynamite.
Batman Begins, he kills the hut full of ninjas by blowing it up and is responsible for Ghul's death.
The Dark Knight, responsible for Harvey Dent's death.