Continuity mistake: An estate car can be seen crossing the background in one scene, as opposed to the village "taxis."
Continuity mistake: When Number 6 writes the coded message he uses black ink, after Number 2 intercepts the message it changes to blue ink. (00:37:24 - 00:38:30)
Continuity mistake: Number 6 gets into the buggy with the hypnotised woman (Number 8, I think) however, when seen from behind, as the vehicle drives away, there is now only one person in the buggy. (00:29:47)
Free for All - S1-E4
Continuity mistake: During the council meeting Number 2's gavel jumps around on his table from shot to shot. (00:16:23)
Do Not Forsake Me Oh My Darling - S1-E13
Continuity mistake: There are many shots of The Colonel (whose mind has been switched with Number 6's mind) driving KAR 120C all around London. The footage is stock, shown in other episodes, and features either Number 6 or his stand in, but definitely not the Colonel.
Free for All - S1-E4
Continuity mistake: When Number 2 and Number 6 are having breakfast, Number 2's ID badge keeps disappearing and re-appearing underneath his scarf.
Chosen answer: We're led to believe Rover is some kind of robot in the series. According to books on the show, Patrick McGoohan was looking for an unusual guardian for the Village, looked up at the sky, and saw a weather balloon, which was what Rover was.
Captain Defenestrator