We prove a version of Montel's Theorem for the case of continuous functions defined over the field $\Bbb Q_p$ of $p$-adic numbers. In particular, we prove that, if $\Delta^{m+1}_{h_0}f(x)=0$ for all $x\in\Bbb Q_p$, and $h_0$ satisfies $|h_0|_p=p^{-N_0}$, then, for all $x_0\in\Bbb Q_p$, the restriction of $f$ over the set $x_0+p^{N_0}\Bbb Z_p$ coincides with a polynomial $p_{x_0}(x)=a_0(x_0)+a_1(x_0)x+\cdots+a_m(x_0)x^m$. Motivated by this result, we compute the general solution of the functional equation with restrictions given by $\Delta^{m+1}_hf(x)=0$ ($x\in X$ and $h\in B_X(R)=\{x\in X:\|x\|\leq r\}$), whenever $f:X\to Y,X$ is an ultrametric normed space over a non-Archimedean valued field $(\Bbb K,|\cdot|)$ of characteristic zero, and $Y$ is a $\Bbb Q$-vector space. By obvious reasons, we call these functions uniformly locally polynomial.