This article is mainly concerned with the time-discretisation of multivalued Hamiltonian systems with multivalued dissipation, a special class of differential inclusions. Two classes of set-valued Hamiltonian systems are considered, depending on the dissipation function being position or momentum dependent. The backward-Euler discretisation is analyzed in both cases: well-posedness of the generalized equation obtained after discretisation is proved, then stability of fixed-points is tackled. The well-known sliding-mode twisting and super-twisting algorithms, as well as an example from Contact Mechanics, illustrate the theoretical developments.