Character mistake: The Enterprise is targeted by projectiles from the planet. Spock identifies them as "plasma torpedoes", and Captain Pike reacts with surprise; "Plasma torpedoes? That's 21st century tech" and from that they all establish that the civilization they contacted is not yet capable of warp drive. Plasma torpedoes are the main weaponry of the Romulans centuries later, and Earth itself did acquire warp drive in the 21st century, so this claim and deduction don't seem to add up. (00:22:15)
Factual error: The bad guy tells the townsfolk to 'Get out of Dodge.' Not a saying used in 1879, despite Dodge City being a real place. In fact by this time Dodge City was no more violent than many other places, so it wouldn't have been used as a direct comparison either. There's no record of this specific phrase being used until Gunsmoke, in 1955.
Continuity mistake: Obi-Wan Kenobi exposes Haja Estree. He asks for his help. When he hastily says "Yeah I'll do it!" Kumail Nanjiani's right hand is curled in a loose fist or fully opened, back and forth as camera angles switch. (00:08:25)