I was listening to a Tribe (Indians) game the other day, and that very thing happened. Apparently they awarded the batter a hit (or that might have been called ball 4, I'm not entirely sure which, but I THINK it was a hit. It's definitely not a strike.
Rule 10:13(f) When an umpire awards the batter or any runner one or more bases because of interference or obstruction, charge the fielder who committed the interference or obstruction with one error, no matter how many bases the batter, or runner or runners, may advance
In a 'catcher interference' call, the batter is awarded first base, and it is not an official 'at-bat', thus not counting in the batter's total 'at-bats' for the game.
On "catcher interference", the batter is awarded first base, no official at-bat is charged (as it is notated in the box score as "'Joe Blow' awarded first base on catcher interference) and the catcher is charged an error.
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No. Nor is it a "time at bat", just like a base on balls, a sacrifice hit, or a sacrifice fly.