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We study the Galois action on paths in the $\mathbb{Q}_\ell$-pro-unipotent\'etale fundamental groupoid of a hyperbolic curve $X$ over a $p$-adic fieldwith $\ell\neq p$. We prove an Oda--Tamagawa-type criterion for the existenceof a Galois-invariant path in terms of the reduction of $X$, as well as ananabelian reconstruction result determining the stable reduction type of $X$ interms of its fundamental groupoid. We give an explicit combinatorialdescription of the non-abelian Kummer map of $X$ in arbitrary depth, and deduceconsequences for the non-abelian Chabauty method for affine hyperbolic curvesand for explicit quadratic Chabauty.

The local theory of unipotent Kummer maps and refined Selmer schemes