Has there ever been a complete game in which the pitcher never threw 3 balls in one at-bat?

Question

During his perfect game in Game 5 of the 1956 World Series, Don Larsen went to 3 balls in a count against only one batter. (source)

Has there ever been a complete game in MLB history in which the pitcher never threw 3 balls during one at-bat? (Meaning that the count against any one batter never reached 3-0, 3-1, or 3-2.)

Answer 1

Per an SI article, Len Barker did not allow three balls to anyone in his 1981 perfect game.

He struck out 11 batters—all swinging—and didn't go to a single three-ball count. Only eight Toronto batters got as much as a two-ball count.