where $\beta ! = \prod_{j = 1}^{d} i_{j}!$.\
We are now ready to gather the results on ultrametric $(C, \alpha)$-good functions that we need. We begin with Theorem $3.2$ from [@KT].

\[theorem 3.2\] Let $V_1,V_2,\cdots,V_3$ be nonempty open sets in F, ultrametric field. Let $ k\in {\mathbb{N}}$, $A_1,\cdots,A_d> 0 $ and $ f\in C^k(V_1\times\cdots,\times V_n) $ be such that $$\label{eqn 3.3}
                |\Phi_j^kf|\equiv A_j \text{ on } \nabla^{k+1}V_j\times\prod_{i\neq j}V_j , j=1,\cdots,d.$$ Then f is $(dk^{3-\frac{1}{k}},\frac{1}{dk})$-good on $V_1\times\cdots,\times V_n$

The following is an ultrametric analogue of Proposition 1 from [@BaBeVe].

\[Calpha\_Prop\] Let $U_\nu$ be an open subset of ${\mathbb{Q}}_\nu ^d,$ ${\mathbf{x}}_0 \in U_\nu$ and let $\mathcal{F}\subset C^l(U)$ be a compact family of functions $f: U\to {\mathbb{Q}}_\nu $ for some $l\geq 2$. Also assume that $$\label{3.4}
                \inf_{f\in\mathcal{F}}\max_{0<|\beta|\leq l} \ |\partial_{\beta}f({\mathbf{x}}_0)|>0.$$

Then there exists a neighbourhood $V_\nu\subset U_\nu$ of ${\mathbf{x}}_0$ and $C, \delta > 0$ satisfying the following property. For any $\Theta\in C^l(U)$ such that $$\label{theta_cond}
                \sup_{{\mathbf{x}}\in U_\nu} \max_{0<|\beta|\leq l} \ |\partial_{\beta}\Theta({\mathbf{x}}_0)|\leq \delta$$ and for any $f\in \mathcal {F}$ we have that

1.  $f+\Theta $ is $(C,\frac{1}{dl})$-good on $V_\nu$.

2.  $|\nabla(f+\Theta)|$ is $ \left(C,\frac{1}{m(l-1)}\right)$-good on $V_\nu$

We follow the proof of [@BaBeVe], which in turn is a modification of the ideas used to establish 