Yes, Batman started out in the first comics carrying a gun and regularly killing criminals by shooting them, strangling them, knocking them off buildings, knocking them into vats of acid (not the Joker, but another dude in the first Batman story), pits, etc.
The rule against killing came later.
Batman "kind of" killed the Joker in the 1989 film Batman, stating his intention by saying outright "I'm going to kill you," and then tying the Joker's leg to a statue that yanked him from a helicopter and sent him plummeting to the ground, dead.
In Batman Returns, Batman overtly kills two people: he turns the batmobile around and blasts the jet engine onto one of Penguin's circus freaks, burning the man alive. Later, he sticks a bomb down the pants of a Strong Man and kicks the guy into a hole where the bomb -- and the man -- blow up. Batman grins at this killing, by the way.
There are other gray areas where Batman can be blamed for maybe round-about "causing" a situation that killed someone, but these are the best examples of Batman outright killing people.
I should note that there are instances of Batman killing people in modern comics history, not merely those early comics. In the mini-series The Cult, for example, Batman is drugged and uses a submachine gun to kill someone. In another 1980s story, he was bitten by the vampire Monk and turned into a vampire temporarily, causing him to kill someone. He has also accidentally killed people, if that counts. Oh, and if we wanted to consider instances where it seems his clear intention was to kill someone -- although it later turned out to not have killed them after all -- we could consider that he tried to trap KGBeast in an underground tomb to die of suffocation/starvation. There are probably a couple of other instances I'm forgetting, too.
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