I think this is a great question. I'm not 100% sure this is a complete correct answer, but I'm at least well over 50% sure.
Here are the paths of the penny (in green) and the train (in red):
First observation: The penny might never have velocity zero; at time $T$ it can jump discontinuously from a positive to a negative velocity.
Second and more important observation: It makes no sense to talk about the "speed of the point of contact" because the point of contact exists only for an instant. To define the velocity at that instant, we'd need to know the point of contact's location in some interval of time around $T$, and there is no such location.
This ignores the (very slight) deviation of the train from a straight red-line course and also ignores all the stuff about deformation of material, all of which is true but none of which (I think) is necessary, because (among other things) this picture seems to answer the question even in a theoretical case where there is no deformation.
Edited to add: It's worth noting that the penny's worldline can't be differentiable. If it were differentiable, it would be tangent to the train's worldline at the point of collision, which means that the penny and the train would share a well-defined velocity at that point. Moreover, that velocity would have to be negative, because the train's velocity is always negative. This would mean that the penny must have turned around (i.e. achieved zero velocity) before the time of impact, which requires you to believe that it saw the train coming and tried to retreat. I hope we can agree to rule that out.
Therefore the picture above is not just one possibility; it's the only possibility (assuming rigid bodies of course).