Radar travels at the speed of light so if your imaginary spaceship is going the speed of light then Houston there is a problem.
No, Houston, it’s no problemo. Even if our ship is going at 0.999999999999999999999999999999999c, any radar beam emitted by it will still pull away from it at c, the full speed of light. We know this because Einstein told us the speed of light is same for all observers, no matter how fast the observer is traveling. Second, our ship is not going to be going the speed of light. Nothing will mass can.