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We study dp-minimal infinite profinite groups that are equipped with auniformly definable fundamental system of open subgroups. We show that thesegroups have an open subgroup $A$ such that either $A$ is a direct product ofcountably many copies of $\mathbb{F}_p$ for some prime $p$, or $A$ is of theform $A \cong \prod_p \mathbb{Z}_p^{\alpha_p} \times A_p$ where $\alpha_p <\omega$ and $A_p$ is a finite abelian $p$-group for each prime $p$. Moreover,we show that if $A$ is of this form, then there is a fundamental system of opensubgroups such that the expansion of $A$ by this family of subgroups isdp-minimal. Our main ingredient is a quantifier elimination result for a classof valued abelian groups. We also apply it to $(\mathbb{Z},+)$ and we show thatif we expand $(\mathbb{Z},+)$ by any chain of subgroups $(B_i)_{i<\omega}$, weobtain a dp-minimal structure. This structure is distal if and only if the sizeof the quotients $B_i/B_{i+1}$ is bounded.

Dp-minimal profinite groups and valuations on the integers