A1 has the ball and has not dribbled. He loses his balance following a fake pass. With pivot foot not moving, A1 uses the ball keep from falling by holding it with both hands and pressing it to the ground.

Does he still have his dribble?

Has he dribbled? No - then he still has his dribble.

Sure he does.

4-15-4 Note 3
It is not a dribble when a player stands still and holds the ball and touches it to the floor once or more than once.

Interestingly enough, had he lifted his pivot foot, he could no longer dribble. But it has has nothing to do with touching the ball to the floor but more to do with the traveling rule.