Praise be to Allah.
If this part of the building is not needed for prayer, then there is nothing wrong with renting it out for permissible purposes that are appropriate to the status and sanctity of the mosque, on condition that the majority of the people of the mosque agree to that. That is based on the ruling that disposing of a waqf and changing its form in ways that will benefit it is permissible.
The Hanbalis regard it as permissible to raise the mosque and build shops beneath it for a purpose. Renting out part of the mosque to cover its expenses comes under the same heading.
In Kashshaaf al-Qinaa‘ (4/375) it says: It is permissible to raise the mosque if most of the people in the location -- i.e., its neighbours -- want that, i.e., raising it and putting shops underneath it which will benefit people. Imam Ahmad stated that according to a report of Abu Dawood, because of the benefits it brings. And the apparent meaning is that it is permissible because of the benefits it brings and the interests it serves. End quote.
See: Majmoo‘ al-Fataawa by Shaykh al-Islam Ibn Taymiyah (31/219); Fataawa al-Shaykh Muhammad ibn Ibraaheem (9/207).
And Allah knows best.
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