I claimed in a previous post that, for a given atomic mass number, there is one and only one nuclide that is immune to all forms of beta decay. It occurs to me that I was probably technically wrong, especially since I made it clear that I considered atoms with different nuclear energy states to be different nuclides.
If a nuclide is beta-stable, I would assume that its excited nuclear energy states (nuclear isomers) also can be, and usually(?) are, beta-stable.
The best example of this phenomenon appears to be hafnium-178 and hafnium-178m2. With a half life of 31 years, hafnium-178m2 has only the ninth longest half life known for a non-ground-state nuclide. But it is the longest-lived one whose ground state is beta-stable. As a bonus, the hafnium-178 ground state is (effectively) stable. So, assuming hafnium-178m2 is in fact beta-stable, this is the pair with the longest minimum half life.