Is there an authentic Hadith from the Prophet, may Allah’s prayers and peace be upon him, that prohibits leaving mosque lights on after prayers, on the grounds that this is a form of extravagance?

No hadith has been reported commanding the extinguishing of mosque lamps. The hadith narrated by Jabir (may Allah be pleased with him) regarding extinguishing lamps upon going to sleep pertains to homes, not mosques, and the reason mentioned therein is not present in modern lamps. It is permissible to keep the mosque lamp lit if there is a benefit, otherwise, it should be extinguished to prevent wastage of money, especially if it is from waqf funds. Scholars have differed on the ruling of keeping the lamp lit all night without need. Some permitted it if there were beneficiaries in the mosque, while others restricted it to what is no more than needed to avoid extravagance or imitating Christians. Some exempted mosques where it is customary to keep them lit all night or if the endower stipulated it. It is permissible to study by the light of the mosque lamp as long as it is within the time of need for prayer, or for no more than one-third of the night. However, if the lamp serves a benefit, such as indicating the mosque's location, then keeping it lit is legitimate to the extent that this benefit is achieved.