I was thinking about this the other day, and I still can't find an answer. Theoretically, let's say there's a room and all the walls are painted perfectly white, meaning it will reflect 100% of all light, same as the ceiling and floor, pretty much everything in this room reflects all light. The room is also a vacuum. If a point of light was placed in the room and removed quickly, would the room ever get dark?
No, the photons produced by the light would, unless you want to break the laws of physics, eventually (actually pretty quickly), give up their momentum to the electrons in the walls as they continued to bounce around. The light would grow less and less intense as the photons lost energy at each bounce, until they were undetectable and the room would be back to how it was before the lamp was lit.
The problem is there is no such thing as perfectly reflecting walls, but if there was you would be correct. It's like after the big bang, we still have microwave light that we can observe from 13B years ago! Due to QM when photons interact with electrons in atoms there is always some probability of absorption, however small.
If the walls and your own body were somehow perfectly reflecting, and the room was a vacuum so that the air doesn't absorb any light, it would just reflect endlessly. The light would quickly become dim because it would disperse throughout the room, rather than coming from a smaller source like a lightbulb, but after limiting to this dimness it would not go dark.
But there is a catch. If you want to be able to see the light, your eye will have to absorb it, and so over time the room will get dimmer.
An interesting question on first blush, yet if you consider what we colloquially mean by "lit up", one imagines being able to see the light of the room, but this seeing necessarily represents a leak of the photons away from the room, as how else can you see the light. So in the overly idealized case of perfectly mirrored walls on all sides in some kind of classical paradise, the light will bounce around ad infinitum, but in any real setting in which you could detect the light, the detection of the light necessitates the energy escaping, and due to the speed of light being rather fast, any small diffusion would build up quite quickly.
A 'perfectly white' room won't always be lit up. The 'always' proviso reminds us of the passage of time, and thermodynamics will ALWAYS triumph given enough time. Even a perfect reflector (there are some internal reflections in glass that are lossless) will have some temperature, and a light wave will interact by doppler-shifting if by no other process.
So, light will, in the fullness of time, come to be thermal-radiation-only, which is the unique light profile that does NOT transfer heat from or to the walls of the hypothetical eternal room. That's the only solution that really stands the test of time.
An incandescent room (too hot for my taste) might always have visible light in it, but no inhabitable enclosure will retain the kind of light our eyes can see.
Yes, by definition, since a perfectly white room means, as you say, perfectly - i.e. mathematically exactly 100% - reflective, any light in it will be reflected around forever and never leave, thus it will always be lit up. The other answers here are basically amounting to that "no such room is possible in practice", which is true, but that is not a real answer since the question presupposes such an impossible, fictive concept.
That said, however, it is also true that no one would be able to see the eternal light as such. For the moment that a person - or even a pair of disembodied eyes - were to be magically teleported into the room, since such necessarily must be less than 100% reflective because in order for the eyes to be able to see any light they must absorb some. Once that happens, the room is effectively now less than perfectly white, and thus all the light will disappear virtually instantaneously from a human point of view, thanks to its fast propagation speed, sucked up by the viewer's eyes (even if every other part of their body could also be somehow made 100% reflective just same).