Infinitesimal deformation of p-adic differential equations on Berkovich curves

We show that if a differential equations $\mathscr{F}$ over a quasi-smooth Berkovich curve $X$ has a certain compatibility condition with respect to an automorphism $\sigma$ of $X$, and if the automorphism is sufficiently close to the identity, then $\mathscr{F}$ acquires a semi-linear action of $\sigma$ (i.e. lifting that on $X$). This generalizes the previous works of Yves Andr\'e, 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.