A fluid pressure measurement sensor (11) comprises a microelectromechanical system (MEMS) chip (23). The MEMS chip (23) comprises two lateral walls (56), a sensitive membrane (49) connected to said lateral walls (56) and sealed cavity (9). The exterior surfaces of the lateral walls (56) and the sensitive membrane (49) are exposed to the fluid pressure. The lateral walls (56) are designed to subject the sensitive membrane (49) to a compression stress transmitted by the opposite lateral walls (56) where said lateral walls (56) are connected to the sensitive membrane (49) such that the sensitive membrane (49) works in compression only. The MEMS chip (23) also comprises a stress detection circuit (31) to measure the compression state of the sensitive membrane (49) which is proportional to the fluid pressure.