The last Greek-ish state to fall to the Ottoman Empire was the Principality of Theodoro, in 1475. You know of it as Gothia: it’s in the Crimea, where Gothic survived to be recorded in the 16th century, before yielding to Greek. The Greek of the Crimea in turn survives as Mariupol Greek.
But the Principality of Theodoro was not a Byzantine outpost; it started as part of the Empire of Trebizond, before becoming autonomous. The Empire of Trebizond, which held out until August 1461, started out claiming to be a successor state of the East Roman Empire, after the Fourth Crusade. The Empire of Nicaea and the Despotate of Epirus did too. But Nicaea was the empire that ended up taking Constantinople back; so its claim was the claim that counted. And, Wikipedia informs me,
In 1282, John II Komnenos stripped off his imperial regalia before the walls of Constantinople before entering to marry Michael’s daughter and accept his legal title of despot. However, his successors used a version of his title, “Emperor and Autocrat of the entire East, of the Iberians and the Perateia” until the Empire’s end in 1461.
So it depends on whether, by Byzantine, you mean “part of the empire calling itself Roman, and whose capital was Constantinople”; “successor state of the pre-1204 Roman Empire whose capital was Constantinople”; or “state whose official language was Greek”. I don’t feel comfortable calling a state Byzantine if it doesn’t have Byzantium.
Now, the Byzantine Empire in 1453 consisted of Constantinople and a bit of the Peloponnese, around Mystras. The Despotate of the Morea was a province of the Byzantine Empire, that remained in Greek hands past 1453, under Demetrios Palaiologos and Thomas Palaiologos, brothers of Constantine XI. Not that they called themselves emperors after 1453, and not that they did much keeping of the Morea in Greek hands: they invited the Ottomans in to subdue a revolt by Albanians, right after the fall of Constantinople: Morea revolt of 1453–54
Mehmed had enough when the Palaiologos brothers started fighting each other. Mystras surrendered to Mehmed the Conqueror in 1460. The very last fortress of the Despotate, Salmeniko Castle, held out until July 1461 under Graitzas Palaiologos.
I don’t want to dignify the post-1453 Despotate of the Morea with the name Byzantine; but unlike Theodoro and Trebizond, there really isn’t anything else to call it. And Thomas Palaiologos was recognised in the West as the rightful heir of the Byzantine Empire.
So, thanks Dimitris Sotiropoulos, for the question: I learned something new today!