There are some weird edge cases in physics where you can move faster than the speed of light. We don't have any means, yet, to move faster than light in a vacuu

There are some weird edge cases in physics where you can move faster than the speed of light. We don't have any means, yet, to move faster than light in a vacuum, but light slows down as it enters materials (0.75c in water, for example). One effect is Cerenkov radiation, which is light emitted when a charged particle moves through a medium faster than light does. You see this in nuclear reactors. It's basically a 'photonic' boom that happens in the wake of the moving charge. Your scenario works for things that are far enough away that the travel time of the photon is significant. Another poster mentioned the Sun. While that example is wrong (the Sun moving), it's true that if the Sun was whisked out of existence, we wouldn't know about it for 8 minutes. An example of this is supernovae in very distant stars. We see the flash of the explosion many years after it happened.