Yes, the ball has to stop moving at a single point in time. This is because its velocity changed directions, from towards the train to away from the train, in the fraction of a second when the ball hit the train.
So why doesn't the train stop even if the ball did (at one point in time)? This situation is a little tricky so we have to analyze it carefully. The short answer is that, at the moment the ball is not moving on the average, the part in contact with the train is moving more or less with the train while the other side is moving toward the train.
Here is the long answer: When a tennis ball hits a hard surface like the front of a train, what probably happens is that the ball presses against the train, while deforming more and more until a maximum, and then the force due to this deformation makes the ball spring back in the opposite direction.
Let's look at these events from the point of view of a man on the train. He sees that the ball rushes to the train at some speed (speed of ball in air + speed of train). Then during contact, it slows down, stops, and then speeds up. Finally it loses contact with the train, and moves in the opposite direction with the same speed. To him, the ball stops exactly when it is maximally deformed (refer to bottom left in diagram), and there is no confusion.
Now let's consider the point of view of a person standing stationary on the ground looking at these events. He sees that the ball rushes to the train. Although the train does not stop, the ball can stop while it touches the train because part of the ball is moving with the train and part is moving the other way (refer to boxed part in diagram). The key difference here is that when the ball stops, relative to him, it has not been maximally deformed yet. So he sees the ball stop earlier than the man on the train sees it stop.
I hope this clears your doubts!
- Mae (+mbw)
(published on 01/26/2011)