Infinitesimal deformation of p-adic differential equations on Berkovich curves

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We show that if a differential equations F over a quasi-smooth Berkovich curve X has a certain compatibility condition with respect to an automorphism σ of X, and if the automorphism is sufficiently close to the identity, then F acquires a semi-linear action of σ (i.e. lifting that on X). This generalizes the previous works of Yves André, Lucia Di Vizio, and the author about p-adic q-difference equations. We also obtain an application to Morita's p-adic Gamma function, and to related values of p-adic L-functions.