Let's say you have the Earth, or any celestial body, spinning as it is. What if you build a tower from the surface, and extended it out into space. If it was built far enough, could the furthest end exceed the speed of light?
What you're asking is essentially whether anything can rotate faster than the speed of light. Just like how it would take infinite energy to accelerate an object to the speed of light in a straight line, it would also take an infinite amount of energy to rotationally accelerate an object to the speed of light.
In any practical sense, this tower would be torn apart by tension long before you would reach luminal speeds, which is actually one of the challenges with building a space elevator.