I'm not sure if excluding Royalty (or royal families) would make a difference or not, but my interest is in both variations.
The family of the Capetians is the oldest verifiable living lineage, the cadet lines of the Capets holding the French monarchy for nearly 1000 years. The Capetian line extend back to late Carolingian period via their forebears Robertians named after Robert the Strong.
The Welfen (Welfs), the family that includes the House of Hanover, claim to be the oldest documented lineage in Europe. They claim to date back to a frankish noble Ruthard who died in 790. German Source.
The Massimo family claim to be descendents of Quintus Fabius Maximus a Roman Consul who fought Hannibal in the 270's BCE. He is mostly famous for giving name to the Fabian strategy. Obviously they cannot prove it but the claim has been around for centuries.
Does "Bishop of Rome" or "Pope" not count?