I have an old classic 4th gen 20gig iPod where the center button stopped working. When trying to press the center button it would not move. No matter how hard I pressed the button it was frozen solid. After opening the iPod the button started working. It turns out the lithium battery in the iPod is right underneath the scroll wheel and over the years the battery swelled to the point where it was jamming the scroll wheel center button.
The most difficult part about replacing the battery is getting the iPod open. I won’t go into the details because there many howto videos to document the procedure. So if you are handy and have some good close-up vision it’s not too hard to disconnect a few ribbon cables to get the battery replaced. Considering the cost of a new iPod classic it’s worth the effort.