There exists a theory of the Schnirelman Shnirelman integral providing Cauchy-type formulas for $\mathbb C_p$-valued rigid (Krasner) analytic functions on subsets of $\mathbb C_p$. For a modern exposition see M. M. Vishik, Non-Archimedean spectral theory, J. Soviet Math. 30 (1985), 2513--2554.
There exists a theory of the Schnirelman integral providing Cauchy-type formulas for $\mathbb C_p$-valued rigid (Krasner) analytic functions on subsets of $\mathbb C_p$. For a modern exposition see M. M. Vishik, Non-Archimedean spectral theory, J. Soviet Math. 30 (1985), 2513--2554.