On the concept of non-ultrametric non-Archimedean analysis

Given some non-Archimedean field $\mathbb{K}$ and some $\mathbb{K}$-linear space $X$, the usual way to define a norm over $X$ involves the {\em ultrametric inequality} $\|x+y\|\leq\max\{\|x\|,\|y\|\}$. In this note we will try to analyse the convenience of considering a wider variety of norms. The main result of the present note is a characterisation of the isometries between finite-dimensional linear spaces over some valued field endowed with the norm $\|\,\cdot\,\|_1$, a result that can be seen as the closest to a Mazur--Ulam Theorem in non-Archimedean analysis.