Not as we know it. No, time did not exist.
However, according to string theory something like it probably did. So pre-bang, a singularity formed that created space time as we know it and started a new beginning.
According to Aztec myth, this is the fifth "time" that has existed. A god sacrificed himself to bring it into reality. (I just did an Aztec hub so that is where my head is, at the moment.) It's as good an answer as anybody can give us, though, at this point in science, or at least as far as I know.