We have studied the mechanical properties of a two-dimensional (2D) boron nanoribbon network (BNRN) subjected to a uniaxial or a biaxial tensile strain using first principles calculations. The results show that the 2D BNRN is super-stretchable. The critical tensile strains of the BNRN in the χ-h1 phase along the a- and b-directions are 0.51 and 0.41, respectively, and that for the biaxial strain reaches an ultrahigh value of 0.84. By analyzing the B-B interatomic distance, coordination number and charge distribution, it is found that with increasing biaxial tensile strain, the χ-h1 BNRN undergoes two structural phase transitions, which are characterized by breaking of the B-B bonds and the partial transformation of the nanoribbon-like structures into chain-like structures. The strain-induced phase transitions significantly reduce the strain energy. We also discuss the elastic constants, Young's modulus, shear modulus, and Poisson's ratios. The super-stretchable and flexible mechanical properties of the BNRNs, together with their superior transport properties, make BNRNs useful in a wide range of applications in nanoscale electronic devices.