Negative linear compressibility (NLC), a rare and important mechanical effect with many application potentials, in a crystal of α-BiB3O6 (BIBO) is comprehensively investigated using first-principles calculations and high-pressure synchrotron X-ray diffraction experiments. The results indicate that the BIBO crystal exhibits the second largest NLC among all known inorganic materials over a broad pressure range. This unusual NLC behaviour is due to the rotation and displacement of the rigid [BO3] and [BO4] building units that result in hinge motion in an umbrella-like topology. More importantly, the parallel-polar lone-pair electrons on the Bi(3+) cations act as "umbrella stands" to withstand the B-O hinges, thus significantly enhancing the NLC effect. BIBO presents a unique example of a "collapsible umbrella" mechanism for achieving NLC, which could be applied to other framework materials with lone-pair electrons.