This was a major conundrum in the early years of understanding the origin and evolution of the spiral arms in galaxies. Due to differential rotation, stars near the center of the spiral orbit the nucleus in only a few million years, while near the outer rim it can take 250 million years. If a spiral arm were a coherent physical entity like one of YOUR arms, its inner parts would get wrapped up into a tight ball of stars while the outer part of the arms is left behind.
Spiral arms are now seen as patterns of star forming activity that sweep around the nucleus of a galaxy at a so-called 'pattern speed'. Gas clouds continuously enter and leave this pattern, and when inside it, are stimulated to form giant molecular clouds and active star forming regions that light-up the entire pattern. This pattern is a gravitational instability wave which naturally forms in any thin, rotating disk of gas that is not stabilized by a large nuclear mass or a large halo.
Spiral arms are not a solid object, but represent a locus IN TIME of star-forming events triggered by a gravitational instability which is itself not a solid object. In fact, if you were to photograph a spiral galaxy using only the red light from its oldest stars, you would not see the spiral pattern at all. Only the light from massive, young stars a few million years old actually reveal the spiral pattern. If you use arm tracers that have an even broader age spread, the arms get more and more washed out.