Over the past few years we have been witnessing a surge of scientific interest to materials exhibiting a rare mechanical effect such as negative linear compressibility (NLC). Here we report on strong NLC found in an ionic molecular crystal of sodium amidoborane (NaAB) - easily-accessible, optically transparent material. In situ Raman measurements revealed abnormal elongation of B-N and N-H bonds of NaAB at pressure about 3 GPa. Ab initio calculations indicate the observed spectroscopic changes are due to an isostructural phase transition accompanied by a stepwise expansion of the crystal along c axis. Analysis of calculated charge density distribution and geometry of molecular species (NH2BH3) univocally points to a chemically driven mechanism of NLC - pressure-induced formation of hydrogen bonds. The new H-bond acts as a "pivot screw" coupling N-H covalent bonds of neighbor molecular species - a system resembling a two-lever "jack device" on a molecular scale. A mechanism based on formation of new bonds stands in apparent contrast to mechanisms so far reported in majority of NLC materials where no significant alteration of chemical bonding was observed. The finding therefore suggests a qualitatively new direction in exploration the field towards rational design of incompressible materials.