Yes it is still running. It came up as the first in my search. AND it's not Alexandria. "The oldest library that is still in operation is the library of St Catherine's Monastery, located in Egypt's Sinai Peninsula at the foot of Mount Sinai (Jebel Musa). The library was established at the time of the construction of the monastery, between 527 and 565 CE, on the orders of Byzantine Emperor Justinian I."But it has continuously been operational. The library of Alexandria was burnt to the ground & rebuild. But there library considered to be the longest operating (but only to the residents) is in a monastery in Syria.