That's true for an ideal "rigid" stick/body, which is just a body for which shocks transfer instantaneously. Reality is that no body is perfectly rigid, obviously, and the shocks propagate depends on th density of the object. Hit a long stick on one side and feel the hit with a delay.

OPs question becomes even more interesting if you start making assumptions about the stick being super light, so really low density, so a normal human could potentially move it. Haven't figured out the answer, a convincing one, though.

I used to think that I had theoretically cracked FTL communication through the use of really long sticks (though the engineering would be its own impossible issue). Took me ages before finding out about compressing objects, which made me feel better because I was sure someone must have already thought of it and disproved it, I just couldn't find the answer until a few years ago!