When light is reflected, I think it is actually absorbed and re-emitted by the surface, so your stipulation that it is not absorbed might need to be adjusted to say that all the light is totally reflected.
It would have to a pretty special conditions type of box; if it was about a meter across the number of reflections for a single ray of light would be around 300 million times per second.
Even if you could construct and isolate a box system that you thought might be able to support the on going reflection for a few minutes or hours, another question pops up... How would you propose verifying it? Any device - your eye, a camera, or other kind of photo-detector will operate by absorbing light. If you did have a box of light, how would you be able to tell?