We study the impact and subsequent retraction dynamics of aqueous liquid droplets upon high-speed impact on hydrophobic surfaces. Often a spectacular "rebound" of the droplet can be observed: after the impact and expansion, the drop retracts rapidly, leading to ejection of part of the material from the surface. We show how non-Newtonian flow properties can be used to slow down the retraction sufficiently to completely inhibit rebound. The slowing down is due to non-Newtonian normal stresses generated near the moving contact line of the droplet. We provide a quantitative theory for the slowing down, and show that the non-Newtonian effects profoundly change the contact line dynamics.