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Dynamic motions are a key feature of robotic arms, enabling them to perform tasks quickly and efficiently. Soft continuum manipulators do not currently consider dynamic parameters when operating in task space. This shortcoming makes existing soft robots slow and limits their ability to deal with external forces, especially during object manipulation. We address this issue by using dynamic operational space control. Our control approach takes into account the dynamic parameters of the 3D continuum arm and introduces new models that enable multi-segment soft manipulators to operate smoothly in task space. Advanced control methods, previously afforded only to rigid robots, are now adapted to soft robots; for example, potential field avoidance was previously only shown for rigid robots and is now extended to soft robots. Using our approach, a soft manipulator can now achieve a variety of tasks that were previously not possible: we evaluate the manipulator's performance in closed-loop controlled experiments such as pick-and-place, obstacle avoidance, throwing objects using an attached soft gripper, and deliberately applying forces to a surface by drawing with a grasped piece of chalk. Besides the newly enabled skills, our approach improves tracking accuracy by 59% and increases speed by a factor of 19.3 compared to state of the art for task space control. With these newfound abilities, soft robots can start to challenge rigid robots in the field of manipulation. Our inherently safe and compliant soft robot moves the future of robotic manipulation towards a cageless setup where humans and robots work in parallel.