A self-sustaining interaction between a roll/streak structure and a three-dimensional Tollmien–Schlichting wave is considered at high-Reynolds-number within the asymptotic suction boundary layer. Strongly nonlinear governing equations, taking the form of a vortex–wave interaction (VWI) are derived and solved numerically. Finite amplitude travelling wave states, bifurcating from the lower branch of the linear neutral curve, are obtained. These states exhibit spanwise focusing, developing steep wall-shear gradients at specific spanwise locations as the wave amplitude rises. A spanwise-local analytic analysis reveals explicitly how the solution gradually loses regularity as the nonlinearity of the VWI system is increased.