We propose an advanced method for controlling the motion of a manipulator robot with strict collision avoidance in dynamic environments, leveraging a velocity damper constraint. Unlike conventional distance-based constraints, which tend to saturate near obstacles to reach optimality, the velocity damper constraint considers both distance and relative velocity, ensuring a safer separation. This constraint is incorporated into a model predictive control framework and enforced as a hard constraint through analytical derivatives supplied to the numerical solver. The approach has been fully implemented on a Franka Emika Panda robot and validated through experimental trials, demonstrating effective collision avoidance during dynamic tasks and robustness to unmodeled disturbances. An efficient open-source implementation along examples are provided here: https://gepettoweb.laas.fr/articles/ haffemayer2025.html.