Mobile manipulators, which are intrinsically redundant when the manipulator and mobile base are moving together, are known for their capabilities to carry out multiple tasks at the same time. This paper presents a whole-body control framework, inspired by legged bio-robots, for a velocity controlled non-holonomic mobile manipulator based on task priority. Control primitives, such as manipulability optimization, trajectory tracking of the end-effector and mobile base, and collision avoidance, are considered in the framework and arranged at different priorities. Lower priority tasks are projected into the null space of control tasks with higher priorities. As a result, lower level tasks are completed without affecting the performance of higher priority tasks. Several experiments are implemented to verify the effectiveness of the proposed controller. The proposed method is proved to be an effective way to solve the whole-body control problem of velocity controlled mobile manipulators.