What happens when the leadoff hitter with a 3-2 count check-swings and is called ball four?

When a batter receives a walk on a 3-2 count, regardless of whether they check-swung or not, they are considered to be a runner who is entitled to all the rights afforded to a player in that situation. The rule allows the batter to overrun first base without the risk of being tagged out.

In this specific scenario, since the batter was awarded ball four, they can legally run to first base and are allowed to overrun it. This rule is designed to protect the runner, as they have been given the opportunity to advance safely after being walked.

This concept is reinforced by the general rules of baseball, which aim to further encourage offensive play by allowing baserunners to take advantage of their opportunities without the penalty of being put out simply for overrunning a base. Hence, the correct interpretation of the situation is that the runner is allowed to overrun first base.