I have the same problem with my Mini - the vanity mirror lights dont work - I am not sure about the reading lights?? I no the light works when you open the door though
Passenger compartment fusebox :-
F14 10A Constant power to interior/ reading lamps
Check the bulbs are all ok, check the above fuse, check power is reaching the bulb holders. Both the LH and RH vanity mirror lights and the Front interior light (incliudes LH / RH map reading light & interior light) are all earthed at E5 (Earth stud) which is on the LH door sill (behind the plastic interior trim I think?) this should be checked for making a good contact. (ie. nut / ring terminals tight with no corrosion).
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