We show that a special stability condition of the associated system of oblique projections (the so-called \ell -paracontractivity) guarantees that the corresponding polyhedral Skorokhod problem in a Hilbert space X is solvable in the space of absolutely continuous functions with values in X . If moreover the oblique projections are transversal, the solution exists and is unique for each continuous input and the Skorokhod map is Lipschitz continuous in both spaces C([0, T];X) and W^{1,1} (0, T; X) . Also, an explicit upper bound for the Lipschitz constant is derived.