  On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative
scalar curvature and vanishing second homology, we establish a sharp
monotonicity formula for the proper $p$-Green function along its level sets for
$1<p<3$. This can be viewed as a generalization of the recent result by
Munteanu-Wang \cite{MunteanuWang2021} in the case of $p=2$. No smoothness
assumption is made on the $p$-Green function when $1<p\leq 2$. Several rigidity
results are also proven.
