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We apply the theory of the radius of convergence of a p-adic connection tothe special case of the direct image of the constant connection via a finitemorphism of compact p-adic curves, smooth in the sense of rigid geometry. Inthe case of an etale covering of curves with good reduction, we get a lowerbound for that radius and obtain a new geometric proof of a variant of thep-adic Rolle theorem of Robert and Berkovich. We take this opportunity toclarify the relation between our notion of radius of convergencand the moreintrinsic one used by Kedlaya

Radius of convergence of p-adic connections and the Berkovich ramification locus