If $p_0$  is **not** a critical point of $f$ then the implicit function theorem  states that,  there exists local coordinates $(x^1,\dotsc, x^n)$,  defined in an open neighborhood $U$ of $p_0$ in $\newcommand{\bR}{\mathbb{R}}$  $\bR^n$ such that, in these coordinates we have ($m=\dim X$)
$$
x^i(p_0)=0,\;\;\forall i,
$$
$$
X=\{ x^{n-m+1}=\cdots =x^n=0\},
$$
$$
f(x^1,\dotsc, x^n)=f(0,\dotsc,0)+x^m.
$$
If you now define the (*non-Euclidean*) box $\newcommand{\ve}{\varepsilon}$
$$
B=\big\{ |x^i|< \ve;\;\;i=1,\dotsc, m\big\}.
$$
In this neighborhood, that is *not* an Euclidean ball, the deformation you seek is obvious.