Light does not stay on seems to be on a timer and
Many fans with remote will have two seperate "lights" buttons on the remote. One is to actually use the lights, the other is a delay light used for when leaving the room when it is dark, that automaticly turns off the lights after a delay. It appears that Hampton bay tried simplifying the light, delay light, and dimmer functions all into a single button on most of their remotes, without offering any sort of "owners manual" to explain the functions and operation of these features.
I tried to do some research for a user's manual, but find it disturbing that Hampton Bay does not feel a need to include usage information in the manual for your model of fan.
I would suggest trying different combinations of double-pressing the light button, pressing and holding, and so forth, until something seems to work. On many fans with wireless remote, when the delay feature is activated, you can almost hear a small audible sound (I'm not sure how to describe it, really) every second or two, while it counts down before turning off. This could be helpful while working out the neccessary procedure, at least to know that it is in the incorrect lighting mode.
Also, most Hamilton Bay remote light switchs look a bit wide, as if they may have two or three button under the rubber button pad. Intuitively, I would guess that the left and right parts of the button would likely be for the dimmer features, so first try pressing, double-pressing, and holding the button in it's middle.
Please let me know if we figure something out, so that it may help others who own similar models of cieling fans. Also, if not too inconvenient, could you post a photo of the remote so that we can figure out what type it is? Hamilton Bay seems to have three different remotes used for Reddington III fans, depending on included features.