An assembly (20) includes a MEMS die (22) having a pressure transducer device (40) formed on a substrate (44) and a cap layer (38). A packaging process (74) entails forming the device (40) on the substrate, creating an aperture (70) through a back side (58) of the substrate underlying a diaphragm (46) of the device (40), and coupling a cap layer (38) to the front side of the substrate overlying the device. A trench (54) is produced extending through both the cap layer and the substrate, and the trench surrounds a cantilevered platform (48) at which the diaphragm resides. The MEMS die is suspended above a substrate (26) so that a clearance space (60) is formed between the cantilevered platform and the substrate. The diaphragm is exposed to an external environment (68) via the aperture, the clearance space, and an external port.