Amazingly, after searching and searching on the Internet, I couldn't find any illuminated mouse (aka LED, lighted, light on...) with a simple on/off switch for the illumination. Obviously, this switch is for the lights only and not for the actual power supply, like USB, that makes the mouse
Taking into account the scenario, what's wrong with this? During the day, it doesn't make sense to have the mouse lighted continuously. I know, there is not much expense in having it on, but still, it is non-necessary. Besides, having the mouse with the lights on during the day definitely can cause some people unnecessary distraction.
P.S.: By the way, there are quite a few illuminated computer keyboards with an on/off switch for the illumination, hence more power to bringing it to the mouse!