Uh, practically everything we saw on TOS in space was a starship in some form or fashion, but the issue is whether they were of the prestigious Star Ship Class in context of TOS' universe.
Backed up by this:
MERIK: He commands not just a spaceship, Proconsul, but a starship. A very special vessel and crew. I tried for such a command.
However it is spelled, it is certain a "starship" - meaning starship class, I imagine - is not anything like your average warp driven spacecraft. The SS Beagle
of the merchant service obviously had warp drive, which makes it a Star Ship, but not a Starship.