If a batted ball hits a base runner in fair territory before the ball has been touched by a defensive player or an umpire, the ball is dead, the runner is called out for interference, the batter is awarded first base, and all other runners advance one base if forced. If, however, the ball has been touched by a defensive player or an umpire before hitting the base runner, the ball is live and play proceeds as usual (unless the runner is judged by the umpire to have deliberately made contact with the ball, then the runner can be called out for interference, the ball is dead, the batter is awarded first base, and all other runners advance one base if forced, or, if flagrant, the batter can also be called out and no runners advance).
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No - the entire plate is in fair territory, so a batted ball that hits the plate is in play; however, if it goes foul after hitting the plate, it is a (foul) dead ball.
The ball is only dead until the batter reaches first base and the pitcher has the ball. After that the batter can steal and the pitcher is free to do what he wants.