The Slowest Gun in the West (1960)

The town of Primrose, Arizona is beset by outlaws, so the towns people hire Fletcher Bissell III (aka The Silver Dollar Kid) as their new sheriff. Fletcher is so cowardly that the townsfolk are sure that the local outlaws will be too proud to gun him down.

This proves to be the case, and the outlaws hire their own cowardly gunfighter, Chicken Farnsworth, to go up against The Silver Dollar Kid.