We use explicit solutions to a drifted Laplace equation in warped
product model spaces as comparison constructions to show
$p$-hyperbolicity of a large class of submanifolds for $p\ge 2$.
The condition for $p$-hyperbolicity is expressed in terms
of upper support functions for the radial sectional curvatures of
the ambient space and for the radial convexity of the submanifold.
In the process of showing $p$-hyperbolicity we also obtain
explicit lower bounds on the $p$-capacity of finite annular
domains of the submanifolds in terms of the drifted $2$-capacity
of the corresponding annuli in the respective comparison spaces.