Let $B$ be the unit ball in $\mathbb{R}^n$. A local version of the Sobolev inequality on $\mathbb{R}^n$ says that for any $p\in[1,\infty]$ there exist constants $C >0$ and $k \in \mathbb{N}$ such that

$$f(0)\leq C\ \sum_{|\alpha|\leq k} \left(\int_B |D^\alpha f|^p\right)^{1/p}$$

for all sufficiently smooth $f$.

Question: Is there a generalization of this inequality to nonarchimedean local fields, and if so, what is the proper interpretation of the derivative in this context.