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He was not Spanish by birth, sure. But he moved to Spain and settled here along with other noble men from the French ultramar families expelled from Cyprus by the Turks. Some of these noble men then went on to New Spain (Mexico) like Bernard among the ranks of the military or as imperial officers.
In Spain people carry two last names, the first from the father and the second from the mother. So in original documents his name appears as BERNARDO DE FLORES Y DE IBELIN