No, according to our current understanding of physics, it is not possible for an elevator to travel faster than the speed of light. The theory of relativity, proposed by Albert Einstein, states that the speed of light in a vacuum is the maximum speed at which information or matter can travel.
Space elevators, while a popular concept in science fiction, are still in the realm of theoretical speculation. A space elevator is a hypothetical structure that connects a point on the Earth's surface to a geostationary orbit above the planet. The idea is to use a cable or tether made of strong, lightweight materials like carbon nanotubes to lift payloads from Earth into space.
The concept of a space elevator relies on the tension in the cable or tether to counteract the force of gravity. It does not involve the elevator itself traveling at high speeds. Instead, the elevator ascends or descends along the stationary cable.
The speed at which the elevator would travel depends on the design and engineering of the system. While it could potentially transport payloads more efficiently and cost-effectively compared to traditional rocket launches, it would still be subject to the laws of physics, including the speed limit imposed by the speed of light.