The tangential branch locus B_X/Y^t∩B_X/Y is the subset of points in the branch locus where the sheaf of relative vector fields T_X/Y fails to be locally free. It was conjectured by Zariski and Lipman that if V/k is a variety over a field k of characteristic 0 and B_V/k^t=∅, then V/k is smooth (= regular). We prove this conjecture when V/k is a locally complete intersection. We prove also that B_V/k^t=∅ implies codim_XB_V/k≤1 in positive characteristic, if V/k is the fibre of a flat morphism satisfying generic smoothness.