The Barbary Corsairs would seem to fit. William Spencer's book 'Algiers in the age of the corsairs' argues that the regency of algiers (a semi independent ottoman vassal state in algeria in which an overclass of turks and europeans ran an arab/berber city with an economy based on piracy and protection money for piracy) was the second most powerful naval power in the med after the british during the aftermath of the french revoloution and the destruction of the french navy. Certainly they were feared to the extent that lots of western powers from italy and iberia through to the low countries, scandinavia and the USA were paying them off to leave their ships alone. The north african corsair cities of Algiers, Tunis, Tripoli and Sale in Morocco were undoutedly major naval powers.

One of the reasons for the english civil war was the devestation they were reaping on english shipping and the idea that charles I wasn't doing enough to protect against them. And Robert Davis estimates nearly 1.5 million white europeans were taken by the corsairs as slaves alone. So undoubtedly effective raiders which indicates a certain ammount of naval power.