Let $G$ be a (finite dimensional real) Lie group, and let $A\subset G$ be an open neighborhood of identity. If $A=\operatorname{Exp}(\mathcal{A})$ is the injective range of the exponential map from a star-shaped neighborhood $\mathcal{A}\subset\mathcal{G}$ of the origin in the Lie algebra $\mathcal{G}$ of $G$, then $A$ is a uniquely divisible neighborhood of identity, i.e., $\exists!\sqrt[m]{x}\in A$ for all $x\in A$ and $m\in\mathbb{N}$.
Question: Given a locally compact Hausdorff topological group $G$, is it always possible to find an open uniquely divisible neighborhood of identity in $G$? If no, then what are the necessary/sufficient conditions?
If $G$ is discrete the question seems to trivialize, so let's assume $G$ is non-discrete. If $G$ is not Lie then I think it needs to have non-trivial subgroups in every neighborhood of identity, but I am not sure this is related.
Thanks in advance.