Wireless Body Area Network (WBAN) consists of health and environmental sensing devices and a dedicated hub node. The hub node, which has more processing power and capabilities, collects and forwards sensed data from the sensing devices to a backend server for further processing and actions. Sensing devices use the CSMA/CA mechanism to communicate with the hub. This access mechanism may cause frame retransmission of corrupted frames due to collisions resulting in performance degradation. The IEEE 802.15.6 standard specifies seven different user priorities for different traffic types. Nodes with higher user priorities have a small channel contention window to access the channel more frequently, causing nodes with lower user priorities to starve. To mitigate this problem, we propose a mechanism based on priority switching. In the proposed mechanism, the nodes with a higher priority switch their contention window to that of the lowest user priority and then switch back to their original contention window after sending a single successful packet. We conducted intensive simulations to validate the proposed mechanism, which shows significant improvement in the packet delivery rate of lower user priority nodes and all the nodes' overall performance in terms of throughput, energy consumption, and inter-frame delay.