The dynamics of magnetic domain walls in ultrathin strip-patterned Pt/Co/AlOx samples with perpendicular easy axis has been studied experimentally under an easy-axis field, superposed to a hard-axis field oriented along the strip. The easy-axis field is large so that the domain walls move well beyond the creep regime. A chiral effect is observed where the domain wall velocity shows a monotonous and surprisingly large variation with an in-plane field. A micromagnetic analysis, combining analytic, one-dimensional, and two-dimensional simulations with structural disorder, shows that this behavior can be reproduced with a Dzyaloshinskii-Moriya interaction of the interfacial type, with due consideration of the dynamics of the tilt degree of freedom of the domain wall. The estimated effective value of this interaction (D≈−2.2mJ/m2 for a 0.6 nm Co thickness) is consistent with values obtained by other techniques. It is also shown, by micromagnetic analysis, that several modes and characteristic times occur in the dynamics of the tilt of such domain walls.