A runner is allowed to overrun 1B 1-way. If he returns to the HP side of 1B, he subject to being tagged out. Ball is live. Abandonment applies when he leaves the baseline {his basepath}, so the call may be delayed until he progresses a reasonable distance away from 1B and does not make an effort to return to base immediately, moving towards a defensive position or the dugout.
When in
doubt, always call em out!