Yes, according to the theory of relativity, a clock on a spaceship traveling at an extremely high speed would be affected. This effect is known as time dilation.
Time dilation occurs because the speed of light is constant in all reference frames, regardless of the relative motion between the observer and the source of light. As an object approaches the speed of light, time appears to slow down for that object from the perspective of an observer at rest.
For example, let's consider a clock on a spaceship traveling at a significant fraction of the speed of light relative to an observer on Earth. From the perspective of the observer on Earth, time would appear to be passing more slowly on the spaceship. This means that the clock on the spaceship would appear to be running slower compared to a clock at rest on Earth.
This phenomenon has been experimentally verified and is an important aspect of the theory of relativity. It has practical implications, particularly for space travel and satellite-based systems like GPS, which need to account for these relativistic effects to maintain accurate timekeeping.