Can a batsman complete a run outside the 2 metre wide cricket pitch?

In hitting a potential winning run in a tied cricket match my brother’s bat slipped out of his hand and over the square leg umpire’s head. He did not realise that a bat is not needed to complete a run, so he ran and collected it then headed for the bowler’s crease, which allowed time for him to be run out.

If he had run parallel to the pitch and reached a point past an imaginary extension of the popping crease at the bowler’s end, would the run of counted, and/or would he have been out?