  Let $p$ be a prime number. We develop a theory of $p$-adic Mahler measure of
polynomials and apply it to the study of $\mathbb{Z}$-covers of rational
homology 3-spheres branched over links. We obtain a $p$-adic analogue of the
asymptotic formula of the torsion homology growth and a balance formula among
the leading coefficient of the Alexander polynomial, the $p$-adic entropy, and
the Iwasawa $\mu_p$-invariant. We also apply the purely $p$-adic theory of
Besser--Deninger to $\mathbb{Z}$-covers of links. In addition, we study the
entropies of profinite cyclic covers of links. We examine various examples
throughout the paper.
