When I switch between spaces, after the transition effect the files on the previous desktop fade back in, so that the same files are on every desktop.
Is it possible to keep separate files on separate Spaces desktops?
It's not; there is only one "Desktop" folder in your home directory, and the contents of that folder determine what appears on all of your desktops.
One problem with allowing varying sets of files for various desktops wouldn't that it's not clear what should happen if you delete a secondary desktop. Where would the files on that desktop go? Lion has made it relatively painless to add and remove desktops, so it's a user experience question that would have to be addressed.
That said, in theory someone could probably write an app that would let you do this. I believe the system sends out notifications when the active desktop changes, so you could probably write an app that watched for those changes and then swapped out the contents of the
~/Desktop/ folder for you. But as far as I'm aware, nobody has written that app yet.