Strings, rods, they're all the same
Yes, you would have instantaneous action at a distance. But such a material can't exist in practice, because on an atomic scale, the vibrations have to travel from one atom to the next. And they are prohibited from doing so any faster than the speed of light, therefore even the stiffest possible theoretical material would only transfer information at (slightly below) the speed of light.

Sorry ragged. No Nobel prize for you this year.
( K2k6 has a proper job these days, Thu 27 May 2010, 8:13, 3 replies, latest was 16 years ago)

Always practical you are.
I want a death star.
( Bartleby A dead man on vacation, Thu 27 May 2010, 8:15, Reply)

I quite like a few of the posts on here:
forums.civfanatics.com/archive/index.php/t-97813.html
( Noeli overtheshoulderboulderholderthingstraplatchboobs, Thu 27 May 2010, 8:23, Reply)

effectively you are just pushing on the atoms at one end
and they in turn transfer that force to the next atoms and so on. There's a ripple that travels down the object to move it. It's just too fast to be observed in objects under about a light-second in length.
( The Light in Chains don't touch the Pope's boner, Thu 27 May 2010, 9:52, Reply)