One micron resolution in a 10mm distance is one part in 10k.
I am presently building a micro-leveller for an inclinometer which will have something like those specifications, for a few GBP.
It starts with a stainless steel M5 threaded rod (studding), driven by a geared stepper motor (28BYJ-48, pocket money on eBay), stepped by an Arduino, running between two ball races, with a brass nut. The rod has a 0.8mm or an 800 micron pitch.
The motor claims it has 2k steps per revolution, so you could theoretically achieve sub micron resolution with just that setup. Well, maybe if you only stepped in one direction. I have measured the backlash and 80 per revolution is a more conservative repeatable accuracy for it.
Turning the rod by 1/80th turn gets 10um resolution at the nut. I gear that down with a lever. In my case it's a compound lever to get 100:1 reduction, giving 100nm resolution. You would only have to gear down 10:1, or take your chances with no levers and fine adjustment in one direction only.
It's very slow, relies for linearity on a cheap bit of threaded rod, and I suspect wear with use will be quite high. But it comes in at the right price.