A p-adaptive continuous residual distribution scheme is proposed in this paper. Under certain conditions, primarily the expression of the total residual on a given element K into residuals on the sub-elements of K and the use of a suitable combination of quadrature formulas, it is possible to change locally the degree of the polynomial approximation of the solution. The discrete solution can then be considered non continuous across the interface of elements of different orders, while the numerical scheme still verifies the hypothesis of the discrete Lax–Wendroff theorem which ensures its convergence to a correct weak solution. We detail the theoretical material and the construction of our p-adaptive method in the frame of a continuous residual distribution scheme. Different test cases for non-linear equations at different flow velocities demonstrate numerically the validity of the theoretical results. © 2017, Springer Science+Business Media New York.