We study the unipotent completion Π un (x0, x1,XK) of the de Rham fundamental groupoid [De] of a smooth algebraic variety over a local non-archimedean field K of characteristic 0. We show that the vector space Π un (x0, x1,XK) carries a certain additional structure. That is a Q p -space Πun(x0, x1,XK) equipped with a σ-semi-linear operator φ, a linear operator N satisfying the relation Nφ = pφN and a weight filtration W• together with a canonical isomorphism Π un (x0, x1,XK) ⊗K K ≃ Πun(x0, x1,XK) ⊗Qur p K. We prove that an analog of the Monodromy Conjecture holds for Πun(x0, x1,XK). As an application, we show that the vector space Π un (x0, x1,XK) possesses a distinguished element. In the other words, given a vector bundle E on XK together with a unipotent integrable connection, we have a canonical isomorphism Ex0 ≃ Ex1 between the fibres. The latter construction is a generalisation of Colmez’s p-adic integration (rkE = 2) and Coleman’s p-adic iterated integrals (XK is a curve with good reduction). In the second part we prove that, if XK0 is a smooth variety over an unramified extension of Qp with good reduction and r ≤ p−1 2 then there is a canonical isomorphism Π r (x0, x1,XK0)⊗ BDR ≃ Π et r (x0, x1,XK0)⊗BDR compatible with the action of Galois group. ( Π DR r (x0, x1, XK0) stands for the level r quotient of Π un (x0, x1,XK)). In particularly, it implies the Crystalline Conjecture for the fundamental group [Shiho] (for r ≤ p−1 2 ) .