This is one of those plays where you can get completely different rulings depending on which rules you're playing under.
One thing in common is that in softball a runner hit by a batted ball while in contact with a base is NOT out- unless he does something to intentionally interfere, like stick out his hand to swat the ball or purposely knock it down.
Now things get different...
As stated, for USSSA the ball remains live. Play on and get what you get.
For NSA, the ball is immediately dead. Award the batter first base. Advance other runners only if forced.
For ASA, if the ball has passed an infielder (other than the pitcher) the ball remains live. If the ball has not passed an infielder, the ball is dead and the batter gets first.
Three different sanctioning bodies...three different rules for the exact same play.