Certainly there is a space race, but it's mostly around Earth and somehow around Moon. I don't see anyone but Mr Musk seriously trying to settle on Mars, so no race there.

OTOH the Starship is also going to be a major player in the Earth and Moon business. Even just on Earth, it's basically capable of landing a few hundred tons of cargo and passengers anywhere on the planet within an hour or so. Or will be, once the landing gear is attached. It may have interesting military consequences.