Ferroelectric tunnel junctions enable a nondestructive readout of the ferroelectric state via a change of resistance induced by switching the ferroelectric polarization. We fabricated submicrometer solid-state ferroelectric tunnel junctions based on a recently discovered polymorph of BiFeO3 with giant axial ratio ("T-phase"). Applying voltage pulses to the junctions leads to the highest resistance changes (OFF/ON ratio >10,000) ever reported with ferroelectric tunnel junctions. Along with the good retention properties, this giant effect reinforces the interest in nonvolatile memories based on ferroelectric tunnel junctions. We also show that the changes in resistance scale with the nucleation and growth of ferroelectric domains in the ultrathin BiFeO3 (imaged by piezoresponse force microscopy), thereby suggesting potential as multilevel memory cells and memristors.