The other day I had a fun thought that we can have p-adic "fractional" derivatives by extending the usual integer order derivatives to p-adic orders in some special cases.
$$\frac{d^n}{dx^n} \exp(kx) = k^n \exp(kx)$$
We can naively extend $n$ to a p-adic integer here, since $k^n$ is well defined for $|k-1|_p < 1$. Additionally, $e^{kx}$ converges for $|x|_p < p^\frac{-1}{p-1}$, so any linear combination of exponentials of this form will also have a well defined "fractional" derivative.
These are some very harsh restrictions, so my question is, is there a larger class of functions or orders of derivatives that we can have p-adic fractional derivatives? Furthermore as a side question, is there any reason to want to do any of this other than pure novelty?