Ginger, the game you’re describing was called “Spud(d)”, at least in my Neighborhood of Kensington, Brooklyn. It was as you described, and if you hit a player, they got a letter, S, then P, then then U, etc, and sometimes we played it with double D if we wanted it to go on longer. First player to get Spud(d), lost, last one who survived won. We also played it with a volleyball/soccer ball.