Finding root using Hensel's Lemma Hensel's Lemma calculates root of a polynomial $\in \mathbb{Z}_p[X]$ but is there any other significance to other branches of mathematics or outside mathematics? Why is finding root of $\mathbb{Z}_p[X]$ important?

For an application, coding theory is a good example. There is a theory for codes (=modules) over rings of the form $\mathbb Z/(p^s)$ which involves lifting polynomials from $\mathbb Z/(p)[x]$ to $\mathbb Z/(p^s)[x]$.
On the other hand, the real numbers don't play a big role in coding theory, since you can't represent them exactly in a computer.
By the way, Hensel's Lemma not only lifts roots, but more generally also factorizations of separable polynomials. Roots correspond to factors of degree $1$, but the factors to be lifted can be of any degree.
