We present a fiber-optic extrinsic Fabry-Perot interferometer pressure sensor based on a nanothick silver diaphragm. The sensing diaphragm, with a thickness measured in a few hundreds of nanometers, is fabricated by the electroless plating method, which provides a simple fabrication process involving a high-quality diaphragm at a low cost. The sensor exhibits a relatively linear response within the pressure variation range of 0-50 kPa, with a high pressure sensitivity of 70.5 nm/kPa. This sensor is expected to have potential applications in the field of highly sensitive pressure sensors.