TITLE: Can we define a derivative on the $p$-adic numbers?
QUESTION [6 upvotes]: Or on any complete valuated field, really. This question came to me when I was thinking about real and complex differentiability, which we can define in a very similar manner: Let $\mathbb F\in\{\mathbb R,\mathbb C\}$, and $V,W$ normed, finite dimensional vector spaces over $\mathbb F$. Also let $f:V\to W$. We call $f$ $\mathbb F$-differentiable at $x_0\in V$ if there is an $\mathbb F$-linear map $\mathrm Df_{x_0}:V\to W$ such that
$$\lim_{x\to x_0}\frac{\Vert f(x)-f(x_0)-\mathrm Df_{x_0}(x-x_0)\Vert}{\Vert x-x_0\Vert}=0.$$
This gives us a very natural way to talk about the real or complex differentiability of a function $\mathbb C\to\mathbb C$, for instance. Just consider $\mathbb C$ a two-dimensional vector space over $\mathbb R$ or a one-dimensional vector space over $\mathbb C$.
I'm wondering wether this can be generalized in an interesting way. This is my idea: We would need a generalization of a norm to general fields, and of course we would still need to work in complete spaces. Norms can be generalized using a valuation: Let $\mathbb F$ be a field. A valuation is a map $\vert\cdot\vert:\mathbb F\to\mathbb R$ satisfying the following for all $a,b\in\mathbb F$:

$\vert a\vert\geq0$ and $\vert a\vert=0$ iff $a=0$.
$\vert ab\vert=\vert a\vert\vert b\vert$.
$\vert a+b\vert\leq\vert a\vert+\vert b\vert$.

Now let $V$ be a vector space over $\mathbb F$. A norm on $V$ is a map $\Vert \cdot\Vert:V\to\mathbb R$ satisfying the following for all $v,w\in V,~a\in \mathbb F$:

$\Vert v\Vert\geq0$ and $\Vert v\Vert=0$ iff $v=0$.
$\Vert av\Vert=\vert a\vert\Vert v\Vert$.
$\Vert v+w\Vert\leq\Vert v\Vert+\Vert w\Vert$.

A norm on $V$ induces a metric $\Vert x-y\Vert$. We can now consider finite-dimensional, normed vector spaces $V,W$ over $F$ which are complete with respect to the induced metric, and are free to define: A function $f:V\to W$ is $\mathbb F$-differentiable in $x_0\in V$ if there exists an $\mathbb F$-linear map $\mathrm Df_{x_0}:V\to W$ such that
$$\lim_{x\to x_0}\frac{\Vert f(x)-f(x_0)-\mathrm Df_{x_0}(x-x_0)\Vert}{\Vert x-x_0\Vert}=0.$$
Now the question is wether such a definition is actually interesting. For instance, finite fields and finite-dimensional vector spaces over them will be discrete, so while they are complete, evaluating the above limit is still not possible. But I imagine infinite fields, like the $p$-adic numbers, complex $p$-adic numbers, or function fields, could make the definition a sensible one. Has such a thing been done before? Are there any obstacles to the well-definedness? Or something that would make the theory of such derivatives uninteresting?

REPLY [3 votes]: There is a short part on derivatives in $\mathbb{Q}_p$ in Fernando Q.Gouvêa's "$p$-adic numbers".
There is also no reason to restrict to finite-dimensional $V,W$. You might be interested in the notion of Fréchet derivatives.