  We implement an iterative numerical method to solve polynomial equations
$f(x)=0$ in the $p$-adic numbers, where $f(x) \in\mathbb{Z}_p[x]$. This method
is a simplified $p$-adic analogue of Jarratt's method for finding roots of
functions over the real numbers. We establish that our method has a higher
order of convergence than J.F.T. Rabago's $p$-adic version of Olver's method
from 2016. Moreover, we weaken the initial conditions in Rabago's method, which
allows us to start the iteration with a multiple root of the congruence $f(x)
\equiv 0 \bmod{p}$.
