This is the lodge of a Dervish order. Originally it belonged to the complex Alaadin Keykubat built, known for its Fluted Minaret. Tekkeli Mehmet Paşa turned it into a lodge in the 14th century. It is believed to have been used as a small mosque also. It is nowadays used as a fine arts gallery. There was an exhibition concerning a former major (Süleyman Evcilmen) of Antalya when I visited in 2012.
More pictures are in the Kaleiçi gallery.