Does FTL exist?

I am confused because many sources I am reading say FTL doesn’t exist. But, at sublight speeds these colonist should be WAAAAAY older than what is described. Assuming that humanity has expanded quite far. I’ll have a colonist who has a backstory that they travel to many different worlds but they’re quite young chronologically. This indicates that FTL does exist or atleast light speed travel. Because even at FTL and/or light speed travel, it would take a long time to travel. Am I correct in this assumption?

Also, how would an Empire as seen in Rimworld even form without it? Obviously the empire was present across the universe. Something like this wouldn’t really be possible without at least light speed travel.

Edit: I feel the need to emphasize that FTL does not mean that there is no need for crypto sleep. Contrary to popular belief, FTL might only be slightly faster than light speed and thus would still take years or decades to travel between the stars.