Batsman leaves his ground
Owner: B Aitken

"The wicket-keeper was not best pleased as a result of the following incident. The striker hit the ball and set off on what looked like a suicidal single. The non-striker practically flew down the pitch, just made his ground, and then fell backwards onto the pitch to avoid being hit by the ball, which was being thrown in. The 'keeper took the ball, whipped off the bails and appealed for a Run out. The non-striker was well out of his ground at that moment. Why was the appeal turned down?"

The appeal failed because, according to your description of events, the non-striker had already made his ground. The only reason he had left his ground after that was to take evasive action to avoid being struck by the ball. This is permitted under Law 38.2 (a).

Read more about Law 38.2 (Run out) at the MCC website