Highly nonlinear bistable current-voltage (I-V) characteristics are necessary in order to realize high density resistive random access memory (ReRAM) devices that are compatible with cross-point stack structures. Up to now, such I-V characteristics have been achieved by introducing complex device structures consisting of selection elements (selectors) and memory elements which are connected in series. In this study, we report bipolar resistive switching (RS) behaviours of nano-crystalline BiFeO3 (BFO) nano-islands grown on Nb-doped SrTiO3 substrates, with large ON/OFF ratio of 4,420. In addition, the BFO nano-islands exhibit asymmetric I-V characteristics with high nonlinearity factor of 1,100 in a low resistance state. Such selector-free RS behaviours are enabled by the mosaic structures and pinned downward ferroelectric polarization in the BFO nano-islands. The high resistance ratio and nonlinearity factor suggest that our BFO nano-islands can be extended to an N × N array of N = 3,740 corresponding to ~10(7) bits. Therefore, our BFO nano-island showing both high resistance ratio and nonlinearity factor offers a simple and promising building block of high density ReRAM.