I was playing devil's advocate here, to some extent, on a slow Sunday afternoon.
I don't disagree with any of the replies, as long as player control is actually lost. However, there are situations in the case book where A1 is not considered to have lost PC, just because B1 touched the ball and it came loose. 4.43.3.A is an example. According to this play, we are to decide if PC was actually lost or if A1 simply allowed the ball to get away. I guess it's similiar to deciding if a player is attmepting to gain control of a fumble when he bats the ball to the floor or has he actually dribbled.
If PC is lost, then he can certainly dribble again.