A C-minimal analogue with a nowhere differentiable power function

Let $\mathbb{C}_p$ be the completion of $\mathbb{Q}_p^{\mathrm{alg}}$. For $\alpha\in\mathbb{C}_p$, let $f_\alpha$ be the function defined on a sufficiently small ball around $0$ by

$$f_\alpha(x)=(1+x)^\alpha=1+\alpha x+\frac{\alpha(\alpha-1)}{2}x^2+\cdots.$$

Let $M\subseteq\mathbb{C}_p$ be an algebraically closed subfield, and let $v$ denote the valuation.

C-minimal analogue conjecture. There is some $\alpha\in\mathbb{C}_p$ and an algebraically closed subfield $M\subseteq\mathbb{C}_p$ closed under $f_\alpha$ such that the structure $(M,+,\cdot,v,f_\alpha)$ is C-minimal and $\alpha\notin M$, so $f_\alpha$ is nowhere differentiable within the structure $M$.

This proposes a C-minimal analogue of the weakly o-minimal construction studied in the paper, combining C-minimality with failure of generic differentiability. The statement is presented as a direction for further research, and no resolution is given here.