n this article we prove exactness of the homotopy sequenceof overconvergentp-adic fundamentalgroups for a smooth and projective morphism in characteristicp. We do so by first proving a correspondingresult for rigid analytic varieties in characteristic 0, following dos Santos [dS15] in the algebraic case. Incharacteristicp, we then proceed by a series of reductions to the case of a liftable family of curves, where wecan apply the rigid analytic result.