First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately:

Let $\varphi:\mathbb{Q}_p\rightarrow\mathbb{C}$.

Canonically, I would define a derivative of such a function in the following way: $\varphi'(x):=\lim\limits_{h\rightarrow 0} \frac{\varphi(x+h)-\varphi(x)}{|h|_p}, $

provided of course that the limit exists. Is this the usual definition of the derivative?

In real functional analysis, there is the notion of Schwartz functions, which are rapidly decreasing functions supported on $\mathbb{R}^n$, i.e. functions, where for multiindices $\alpha,\beta$

$\sup_{x\in\mathbb{R}^n}|x^\alpha D^\beta (f)(x)|<\infty.$

To my knowledge, the term Bruhat-Schwartz function generalizes this to the case where the domain is a locally compact abelian group, e.g. the field of $p$-adic numbers. Thus there should be an analogue notion where one considers $p$-adic derivatives.

Now I found in some sources, that a Bruhat-Schwartz function on $\mathbb{Q}_p^n$ is a locally constant function with compact support. Unfortunately, I don't see at all how you get from the one definition to the other.