In this work, we have developed a novel bond-associated non-ordinary state-based peridynamic (BA-NOSB PD) model for the numerical modeling and prediction of the impact response and fracture damage of quasi-brittle materials. First, the improved Johnson-Holmquist (JH2) constitutive relationship is implemented in the framework of BA-NOSB PD theory to describe the nonlinear material response, which also helps to eliminate the zero-energy mode. Afterwards, the volumetric strain in the equation of state is redefined by the introduction of the bond-associated deformation gradient, which can effectively improve the stability and accuracy of the material model. Then, a new general bond-breaking criterion is proposed in the BA-NOSB PD model, which is capable of covering various failure modes of quasi-brittle materials, including the tensile-shear failure that is not commonly considered in the literature. Subsequently, a practical bond-breaking strategy and its computational implementation are presented and discussed by means of energy convergence. Finally, the proposed model is verified by two benchmark numerical examples and demonstrated by the numerical simulation of edge-on impact and normal impact experiments on ceramics. The comparison between our results and references shows good capability and stability for impact problems of quasi-brittle materials. Numerical oscillations and unphysical deformation modes are effectively eliminated, showing strong robustness and bright prospects for relevant applications.