No, even the best mirrors absorb part of the light that hits it. Light travels so quickly, that it bounces a hundred times off the mirrors and ends up completely absorbed in a fraction of a second, too quickly for humans to notice.
This, I like this question a lot :) The answer would be YES, but the illumination would be completely undetectable to the human eye, instantly. After all, even light is not instantaneous. But you wouldn't be able to see the light because the light would be 'on' in reflections so far away from you that it wouldn't make a difference, if that makes any sense.
My initial guess is No. But brings me to another question I'm not sure about. If you left the light on, would it make the room brighter since the light is being reflected? Kinda like when you look at snow on a sunny day. Its brighter. Of course the albedo of snow and mirrors are different.