Yes, astronauts do experience time dilation during space travel, albeit to a very small extent. The effects of time dilation experienced by astronauts in space are due to their relative motion and the difference in gravitational fields compared to observers on Earth.
When astronauts are in orbit around the Earth or on long-duration space missions, their velocities are significantly higher than those on the Earth's surface. According to the principles of special relativity, as an object approaches the speed of light, time dilation occurs, causing time to pass more slowly for the moving object compared to a stationary observer.
However, the time dilation experienced by astronauts in low Earth orbit or on space missions is relatively minimal. The velocities involved in these scenarios are not close enough to the speed of light for time dilation to have a noticeable impact on their aging or daily lives. The differences in time dilation between Earth and astronauts in space are measured in microseconds or even less.
On the other hand, in scenarios where astronauts are subjected to much higher velocities, such as future interstellar travel at a significant fraction of the speed of light, the effects of time dilation would become more significant. In such cases, astronauts would indeed experience noticeable time dilation effects, resulting in a discrepancy between their aging and the aging of observers on Earth.
To summarize, while astronauts do experience time dilation to a small degree during space travel, the effects are typically negligible for current space missions. The significant time dilation effects associated with high speeds near the speed of light are not currently achievable with our existing technology.