Yes, according to the theory of special relativity, a clock on a spaceship traveling at an extremely high speed would be affected. Special relativity introduces the concept of time dilation, which means that time can appear to run differently for objects that are moving relative to each other at different speeds.
According to special relativity, as an object moves faster and approaches the speed of light, time slows down for that object relative to a stationary observer. This effect is known as time dilation. It means that a clock on the spaceship would appear to run slower from the perspective of an observer at rest relative to the spaceship.
This phenomenon has been experimentally verified using high-speed particles in particle accelerators. For example, muons, which are subatomic particles with a very short lifetime, can reach the Earth's surface despite their short lifetimes because time dilation allows them to "live" longer from their perspective.
So, if a clock on a spaceship were traveling at an extremely high speed, it would tick slower compared to a clock at rest relative to an observer outside the spaceship. This time dilation effect becomes more significant as the spaceship's speed approaches the speed of light.
It's worth noting that time dilation is a relative effect. Both the observers on the spaceship and the observers outside the spaceship would perceive their own clocks to be running normally, but they would observe the other's clock to be running slower. This is a consequence of the relativity of simultaneity and the fact that different observers can have different measures of time intervals due to their relative motion.