
Nope! Batman will beat criminals up and put them in the hospital with several broken bones and severe injuries but will generally not kill. It is what makes his conflict with the joker so juicy. In the heath ledger movie the joker literally dares batman to kill him as that would be a breaking of his moral code and so the joker would win.
If you are curious then you can search google "does batman kill"

His no kill rule only really extends to major villains hence it doesn't really uphold anyways. Its a running gag in the dc fan area that gotham is only upheld because Batman is known to be so ruthless. You can mess up slightly and be put on a tshirt while the Joker can exterminate half the universe and just be sent to Arkham.
A story where the mc sets out to be both the judge and the executioner, huh?
Batman would not approve.