The OP has a 2001 X5 which has external wiring for the Xenon lights. Check the ballast under the headlight by swapping faulty side to the other side and see if the fault follows it or search fleabay for an OEM replacement and change it out.
If it is not the ballast then the the wiring between the LKM and ballast or the LKM are most likely faulty.
I suggested the OEM ballast because of the rubber boot the comes mounted on the high voltage wire to the Xenon bulb that seals the back of the headlight unit.

__________________
Sent via Telegraph Wires using my Morse Key.
2021 X5 xDrive40i
2015 X5 xDrive35i MSport
(Traded) 2013 X5 xDrive35i