Perfectly rigid bodies are not possible in relativity, although this is not directly related the Lorentz contraction mentioned in the question. One immediate consequences of relativity is that no signal can travel faster than the speed of light; and this actually rules out perfectly rigid bodies.
The reason, although it may not be instantly obvious, is actually fairly simple. If we had a long (length $L$), perfectly rigid rod and apply a force to it, it would need to accelerate uniformly. Perfect rigidity would mean that both ends need to be moving exactly in synchronization; as soon as a force is applied at $x=0$, the other end at $x=L$ has to start to move. (If the don't move together, then the length of the rod has changed.) However, it is impossible in relativity for the far end to start moving at the same time, because that would require a signal to travel instantly down the length of the rod. In actuality, when the force is applied at one end, the rod will deform slightly, and the deformation will propagate at speed $v$ ($v$ is the sound speed in the material, and $v<c$) down the length of the rod. Only after a time $L/v$, when the signal reaches the other end, will the far end start to move.