Me thinks I'll go the route of pure evil, mwahaha
The zombies are actually time-traveling zombies from the future that create a paradox by biting themeslves, thus forcing themselves to develop the time travel technology in the future to go back, instigate continuous loop.
Either that, or these zombies are, quite simply, made from animated skeletons that bite people instead, which, given enough time, turn into said skeletons. You can't kill em, even the bone powder of said skeletons comes after you and freakin chokes you to death. They only way to survive? Not sure yet, but I'm sure it would be something completely silly just to make this no-win scenario kinda, well, winnable. Like maybe these skeletons are scared of cats, or janitors, or brooms...