Let $K =(K,| \cdot |)$ be a non-Archimedean valued field.
Let $E$ be a $K$-vector space. A **norm** on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that:


1. $||x||=0$ if and only if $x=0$,
2. $||\lambda x||=|\lambda|\,||x||$,
3. $||x+y||\leq\max\{||x||,||y||\}$,


for all $x,y\in E$.


A function $||\cdot||:E\to[0,\infty)$ satisfying conditions 1, 2 and $||x+y||\leq ||x||+||y||$ is called an **A-norm** on E.


Clearly every norm is an A-norm, but not conversely.
Why in the literature (van-Rooij, Schikhof, etc,) the study of Banach spaces and the development of non-Archimedean functional Analysis is done with norms instead of A-norms? Does it worth to generalize the results of norms to A-norms? 



Norm vs A-norm in non-Archimedean Functional Analysis
I guess that putting an archimedean norm on a vector space over a nonarchimedean field gives just an uncorrelated product of something archimedean with something nonarchimedean. Number theorists sometimes look at *all* places of $\mathbb{Q}$ at once: all p-adic valuations and the archimedean valuation. I don't see a reason why one should want to look at just two places. I do not expect it would generate any new theory.


Anyway, the reason why people in nonarchimedean functional analysis work with nonarchimedean norms is that the vector spaces over a nonarchimedean field which turn up naturally, come with a nonarchimedean norm. Like finite dimensional vector spaces or spaces of functions.


