We propose and demonstrate a pressure sensor based on a micro air bubble at the end facet of a single mode fiber fusion spliced with a silica tube. When immersed into the liquid such as water, the air bubble essentially acts as a Fabry-Pérot interferometer cavity. Such a cavity can be compressed by the environmental pressure and the sensitivity obtained is >1000 nm/kPa, at least one order of magnitude higher than that of the diaphragm-based fiber-tip sensors reported so far. The compressible Fabry-Pérot interferometer cavity developed is expected to have potential applications in highly sensitive pressure and/or acoustic sensing.