Nope. Imageine instead of ping-pong balls you had a blue squash ball (TIL: the color is important! [0]). They're squishy, and so when you poke the ball at your end, in reality you're causing it to visibly deform. It then pushes back out to a sphere, moving away from you. In doing so, it pushed into the next ball. This process of deform->reform->...deform->reform takes time, and that time is the propagation delay.

[0]: http://en.wikipedia.org/wiki/Squash_(sport)#Playing_equipmen...

No signal can travel faster than light. If you put pressure sensors along the infinitely long tube, you'll see that the signal reaches the points at different times.

We're just so used to seeing it happen instantaneously that it's slightly difficult to imagine otherwise

The signal would only travel at the speed of sound in ping-pong balls. What that is precisely is very difficult to estimate. Try "wave propagation in granular materials" for a Googleable phrase. Even the case of a perfectly aligned 1-D chain of ideal ping-pong balls is not straightforward to analyze; the confinement force probably has a significant effect. (disclaimer: not a physicist)

You're forgetting about propagation of force within the ping-pong ball. It is not immediate.