It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. [Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions](https://arxiv.org/abs/math/0407142) and [Cherry - Lectures on Non-Archimedean Function Theory](https://arxiv.org/abs/0909.4509). I am mainly working on non-Archimedean functional analysis right now and need to become better acquainted with the non-Archimedean analogues of basic results that might be encountered in a first course on complex analysis, up to and including Liouville's Theorem e.g. Cauchy integral formulas, holomorphic functions etc. for a few spectral theory proofs.


Of course I am well aware that with many results there will be no such analogue. What I would like to know is if there exists a good introduction to this area that I could look at, that starts with the fundamentals. For example, is there an analogue of 'holomorphic iff analytic'? Any advice much appreciated.


This relates to my earlier question: [Non-emptiness of spectrum σ(a) in non-Archimedean Banach algebras](https://mathoverflow.net/questions/412631/non-emptiness-of-spectrum-sigmaa-in-non-archimedean-banach-algebras)



Literature on non-Archimedean analogues of basic complex analysis results
Benedetto has a textbook that discusses basic $p$-adic analysis, although his aim is to study $p$-adic dynamics. And it's for a single variable. But might be a good place to get some information.


[Dynamics in One Non-Archimedean Variable, Robert L. Benedetto, Graduate Studies in Mathematics, Volume 198, 2019, American Mathematical Society](https://bookstore.ams.org/gsm-198/)


I'll also mention that these days a lot of $p$-adic analysis is done on Berkovich space, rather than on $\mathbb Q\_p$ or $\mathbb C\_p$. An introduction to analysis on the Berkovich line can be found in the book of Baker and Rumely.


[Potential Theory and Dynamics on the Berkovich Projective Line,
Matthew Baker, Robert Rumely, Mathematical Surveys and Monographs
Volume 159, 2010, American Mathematical Society.](https://bookstore.ams.org/surv-159)


