If light ran in truly mathematical rays, there would indeed be a paradox. It doesn't, it runs in "bundles", called photons, which have a finite size, about the size of the light's wavelength. The bottom of the bundle hits the mirror first, and during the time that the top of the bundle is moving towards the mirror (at C, of course) the mirror itself is moving.
The mirror moves by the exact amount required to change to slope of the mirror to reflect the whole bundle to the proper place directly above the mirror, irrespective of the velocity of the room (lab? spaceship?) holding the mirror.

In the tradition of all lazy textbook writers, the demonstration of this phenomena is left as an exercise to the energetic student.

I can also email you a C program that calculates the Classic Lorentz contraction for any given mirror speed, what it must be for the light to hit the ceiling directly above the mirror, and what the contraction added to the mirror speed actually give.