No, a gravity well cannot cause an object to move faster than light. According to our current understanding of physics, as described by Einstein's theory of relativity, the speed of light in a vacuum is an absolute speed limit. Nothing with mass can travel at or exceed the speed of light.
Gravity wells, such as those created by massive objects like stars or black holes, can certainly affect the motion of objects and bend the path of light through gravitational lensing. However, they do not enable objects to exceed the speed of light.
In fact, gravity itself is limited by the speed of light. Gravitational interactions propagate at the speed of light, meaning any changes in the gravitational field caused by a massive object's movement or changes take time to propagate to distant objects.
So, while gravity wells can have significant effects on objects and space-time, they do not provide a mechanism for achieving faster-than-light travel.