I have an mains powered old FM radio unit with a integrated light. The radio seems to work fine and picked up quite a few stations, although you need to push quite firmly on the "scan" button to tune it. (It has no presets, the only way to change the station is to press "scan").
The unit also has an inbuilt light HOWEVER the light is incredibly weak (it is a low voltage incandescent bulb, not an LED). Also the button to operate the light has broken off, exposing the electric circuit board underneath (see pictures). For this reason I would strongly recommend ignoring the light altogether (maybe putting some tape over the broken button).
It's an incredibly cheaply built thing, but it does still work. Maybe if you wanted a set for a garage / etc. it would be useful.
Mains power, not battery.