Let $R$ be a commutative ring. Is there an example of a free module of non-finite rank over a commutative ring with bases of different cardinalities?
I know the theorem for the finitely generated free module case that any two bases have the same cardinality.
No, two free $R$-modules are isomorphic if and only if the cardinalities of their bases (i.e. their ranks) coincide. That also holds for non-finite cardinalities. In particular, every free basis has the same cardinality, such that one has a well-defined rank.
You can still have two non-isomorphic free $R$-modules of infinite rank by choosing two different infinite cardinals.