See buckyballs as a trivial refutation of your point.
It doesn't break down at all. Diamonds are organic structures, in the classical definition of that term. (Assuming they weren't treated with insecticides in the mine... /s)
It's very much nitpicking and an edge case but now you've got me wondering if some silicone hydrocarbon analogs might not qualify. Noting that we have plausible theories about the feasibility of silicone based life.
... Do we? Last I heard those weren't really viable due to some combination of being too stable or too unstable in all the available solvents.