Not yet reviewed by Pith; the record is open.

We propose a new method for self-injection of high-quality electron bunches in the plasma wakefield structure in the blowout regime utilizing a "flying focus" produced by a drive beam with an energy chirp. In a flying focus the speed of the density centroid of the drive bunch can be superluminal or subluminal by utilizing the chromatic dependence of the focusing optics. We first derive the focal velocity and the characteristic length of the focal spot in terms of the focal length and an energy chirp. We then demonstrate using multidimensional particle-in-cell simulations that a wake driven by a superluminally propagating flying focus of an electron beam can generate GeV-level electron bunches with ultralow normalized slice emittance ($\sim$30 nm rad), high current ($\sim$ 17 kA), low slice energy-spread ($\sim$0.1%) and therefore high normalized brightness ($>10^{19}$ A/rad$^2$/m$^2$) in a plasma of density $\sim10^{19}$ cm$^{-3}$. The injection process is highly controllable and tunable by changing the focal velocity and shaping the drive beam current. Near-term experiments at FACET II where the capabilities to generate tens of kA, <10 fs drivers are planned, could potentially produce beams with brightness near $10^{20}$ A/rad$^2$/m$^2$.