Suppose we have a single variable polynomial $f\in \mathbb{Z}_p\subset \mathbb{Q}_p$ and we have its root $\alpha$ over the finite fields $\mathbb{F}_p$, $f(\alpha)=0$ mod $p$. By Hensel's Lemma we can lift the solution to a solution in $\mathbb{Z}_p\subset \mathbb{Q}_p$. The multivariate Hensel's lemma that I checked is stated in the following PDFs.
https://kconrad.math.uconn.edu/blurbs/gradnumthy/multivarhensel.pdf, http://people.math.harvard.edu/~yifei/Hensel_lemma.pdf
The thing that I want to solve now is, suppose we have an multivariate polynomial equation over the finite fields $\mathbb{F}_p$ with $\bar{\alpha}=(\alpha_1,...,\alpha_n)$ as the root, $f(\bar{\alpha})=0$ mod $p$. Can we lift this to a solution in $\mathbb{Z}_p\subset \mathbb{Q}_p$? I am not getting how to use the multivariate Hensel's Lemma to solve this problem. Can anyone help?