Yes, the Papal States (the state that the Pope ruled over between ~800 and 1870) were to a large extent the successor to the Exarchate of Ravenna, the area that the Byzantine Empire reconquered in Italy from the mid-500s to the mid-700s. Ravenna had become the capital of the Western Roman Empire long before the fall of the Western Roman Empire, since it was closer to the action on the frontiers of Central and Eastern Europe.