Below $X$ is a totally disconnected locally compact Hausdorff abelian group.
If $X$ is $\mathbb Z$ with discrete topology then the only compact subgroup of $X$ is the zero subgroup. So I suggest to interpret the largeness of the searched compact group not as “non-zero”, but as open.
If $\mathcal L$ in Proposition 3.3.9 from [DPS] denotes the class of all locally compact Hausdorff abelian groups then $X$ has a linear topology and contains an open compact subgroup.
An abelian totally disconnected compact (even countably compact) Hausdorff topological group $X$ is reduced, that is $X$ has no non-zero divisible subgroups [AT, Pr. 9.12.A].
Proposition 3.5.9 from [DPS] describes the structure of totally disconnected compact abelian groups.
At last I remark that the group $\mathbb Z_p$ of p-adic integers should be torsion free and $n$-divisible for each $n$ coprime with $p$ (see, for instance, [M, p.4]).
[AT] Alexander V. Arhangel'skii, Mikhail G. Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.
[DPS] Dikran N. Dikranjan, Ivan R. Prodanov, Luchezar N. Stoyanov. Topological Groups, Marcel Dekker, New-York, 1990.
[M] David A. Madore, A ﬁrst introduction to p-adic numbers.