Did the Indians play any games that were like foodball or baseball?
The Indians played many "Field" games. The "balls" were made of natural material, stuffed animal skins, natural wooden knots, polished stones, hemp etc. Shinny games in which the ball or puck is moved with a stick were quite common.There were many variations but these were team games (with three to ten players, typically) with rules and an object (to get the ball past the other team, through posts or into a hole.
There is a terrific book Grass Games & Moon Races by Jeannine Gendar that describes the games and toys of the California Indians. The author takes great care to show how the games fit into the Inidan culture and varied by tribe.