Strontium titanate (SrTiO3) is the "silicon" in the emerging field of oxide electronics. While bulk properties of this material have been studied for decades, new unexpected phenomena have recently been discovered at the nanoscale, when SrTiO3 forms an ultrathin film or an atomically sharp interface with other materials. One of the striking discoveries is room-temperature ferroelectricity in strain-free ultrathin films of SrTiO3 driven by the TiSr antisite defects, which generate a local dipole moment polarizing the surrounding nanoregion. Here, we demonstrate that these polar defects are not only responsible for ferroelectricity, but also propel the appearance of highly conductive channels, "hot spots", in the ultrathin SrTiO3 films. Using a combination of scanning probe microscopy experimental studies and theoretical modeling, we show that the hot spots emerge due to resonant tunneling through localized electronic states created by the polar defects and that the tunneling conductance of the hot spots is controlled by ferroelectric polarization. Our finding of the polarization-controlled defect-assisted tunneling reveals a new mechanism of resistive switching in oxide heterostructures and may have technological implications for ferroelectric tunnel junctions. It is also shown that the conductivity of the hot spots can be modulated by mechanical stress, opening a possibility for development of conceptually new electronic devices with mechanically tunable resistive states.