The Pioneer space probes have a heck of a track record. Pioneer 6 is
probably still working 42 years after liftoff, though they haven't bothered
to contact it since 2000.

Pioneer 11 lasted until late 1995 and Pioneer 10 made it to 2003, at
least in terms of us being able to pick up its signal. They were launched
in 1973 and 1972, respectively, made of mostly 1960s hardware.

The Voyager probes are now over 30 years old and will last until 2025.

I read somewhere that one of the old US Navy Transit satellites launched
in 1964 was still transmitting, but I cannot remember the details.

As for just existing, Vanguard 1 has that distinction. Launched in 1958,
it has an orbital lifetime of 200 years, assuming nobody picks it up beforehand.