Answer
Rooms which are inside the mosque and whose doors open into the mosque come under the same ruling as the mosque. Based on that, it is permissible to observe i'tikaaf in them because they are part of the mosque.
But if they are outside the mosque, then i’tikaaf observed in them is not valid, even if the room has a door leading into the mosque.
Source: Fataawa al-Lajnah al-Daa’imah, 10/411
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