Are there a polynomials $f_1,...f_n \in \mathbb{Z}_p[x_1,...x_n]$ with there coeficients $p$-adic integers s.t.

A map $F:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p^n$ defined by $f_1,...f_n$ satisfy the following property.

1. $F(\mathbb{Z}_p^n)\subset B((0,...,0),1)$ where $B((0,...,0),1)$ is open ball centered at the origin with radius $1$

2. $F$ induces an isomorphism onto $B((0,...,0),1)$ on each open set $B((i_1,...,i_n),1)$ where each $i_1,...,i_n$ ranges from $0$ to $p-1$.

Note that $\mathbb{Z}_p^n=\bigcup_{i_1,...,i_n}B((i_1,...,i_n),1)$