Recently, Wireless Body Area Network (WBAN) has witnessed significant attentions in research and product development due to the growing number of sensor-based applications in healthcare domain. Design of efficient and effective Medium Access Control (MAC) protocol is one of the fundamental research themes in WBAN. Static on-demand slot allocation to patient data is the main approach adopted in the design of MAC protocol in literature, without considering the type of patient data specifically the level of severity on patient data. This leads to the degradation of the performance of MAC protocols considering effectiveness and traffic adjustability in realistic medical environments. In this context, this paper proposes a Traffic Priority-Aware MAC (TraPy-MAC) protocol for WBAN. It classifies patient data into emergency and non-emergency categories based on the severity of patient data. The threshold value aided classification considers a number of parameters including type of sensor, body placement location, and data transmission time for allocating dedicated slots patient data. Emergency data are not required to carry out contention and slots are allocated by giving the due importance to threshold value of vital sign data. The contention for slots is made efficient in case of non-emergency data considering threshold value in slot allocation. Moreover, the slot allocation to emergency and non-emergency data are performed parallel resulting in performance gain in channel assignment. Two algorithms namely, Detection of Severity on Vital Sign data (DSVS), and ETS Slots allocation based on the Severity on Vital Sign (ETS-SVS) are developed for calculating threshold value and resolving the conflicts of channel assignment, respectively. Simulations are performed in ns2 and results are compared with the state-of-the-art MAC techniques. Analysis of results attests the benefit of TraPy-MAC in comparison with the state-of-the-art MAC in channel assignment in realistic medical environments.