The oldest flying 747 is also the fifth 747 produced. It was delivered in August 1970 and is still flying today.
This is true. As the linked article notes, a remarkable number of early 747s are still in service in Iran, either with the Iranian Air Force or as part of the fleet of Iran Air. These airframes date to before the 1979 Iranian Revolution. I didn't really want to open that can of worms, though--given the diplomatic situation between Iran and the United States, one wonders at the level of support that Boeing would feel it had to provide (or even would be allowed to provide, as a U.S. company).
I would wonder if there were any parallels to the situation in, say, Cuba, with its large population of Batista-era (pre-Castro, pre-Communist revolution) U.S.-built "classic" automobiles. Though you'll still see them on the roads of Havana, I wouldn't expect Ford to still have parts for, say, the 1957 Fairlane.