Layered iridates have been the subject of intense scrutiny on account of their unusually strong spin-orbit coupling, which opens up a narrow bandgap in a material that would otherwise be a metal. This insulating state is very sensitive to external perturbations. Here, we show that vertical compression at the nanoscale, delivered using the tip of a standard scanning probe microscope, is capable of inducing a five orders of magnitude change in the room temperature resistivity of Sr2IrO4. The extreme sensitivity of the electronic structure to anisotropic deformations opens up a new angle of interest on this material, with the giant and fully reversible perpendicular piezoresistance rendering iridates as promising materials for room temperature piezotronic devices.