It is a well-known result by Larson and Sweedler that, for finite-dimensional Hopf algebras over a field, the antipode is always a linear isomorphism.
My question is whether this property still holds for free, finite rank Hopf algebras over arbitrary commutative rings.
Looking at the proof in [EGNO, Proposition 5.3.15], it is argued for the field case as follows: for a Hopf algebra $H$ with antipode $S$, one considers the collection of subbialgebras $H_n := \mathrm{Im}(S^n)$. Then by finite-dimensionality, there exists the smallest $n$ such that $H_n = H_{n+1}$. However I think that for Hopf algebras over general rings this argument does not work, as one would have to consider at least a PID, right? Perhaps there are other arguments out there that avoid this issue, but I am not aware of them.
The answer is affirmative. The case for principal ideal domains is a classical theorem by Larson and Sweedler, see Proposition 4 in
Larson, Richard Gustavus; Sweedler, Moss Eisenberg. An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91 (1969), 75--94. MR0240169
The case for general rings was proved later by Pareigis, see Proposition 4 in
Pareigis, Bodo. When Hopf algebras are Frobenius algebras. J. Algebra 18 (1971), 588--596.