  In this article we prove exactness of the homotopy sequence of overconvergent
$p$-adic fundamental groups for a smooth and projective morphism in
characteristic $p$. We do so by first proving a corresponding result for rigid
analytic varieties in characteristic $0$, following dos Santos in the algebraic
case. In characteristic $p$, we then proceed by a series of reductions to the
case of a liftable family of curves, where we can apply the rigid analytic
result.
