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We propose a new method for self-injection of high-quality electron bunchesin the plasma wakefield structure in the blowout regime utilizing a "flyingfocus" produced by a drive beam with an energy chirp. In a flying focus thespeed of the density centroid of the drive bunch can be superluminal orsubluminal by utilizing the chromatic dependence of the focusing optics. Wefirst derive the focal velocity and the characteristic length of the focal spotin terms of the focal length and an energy chirp. We then demonstrate usingmultidimensional particle-in-cell simulations that a wake driven by asuperluminally propagating flying focus of an electron beam can generateGeV-level electron bunches with ultralow normalized slice emittance ($\sim$30nm rad), high current ($\sim$ 17 kA), low slice energy-spread ($\sim$0.1%) andtherefore high normalized brightness ($>10^{19}$ A/rad$^2$/m$^2$) in a plasmaof density $\sim10^{19}$ cm$^{-3}$. The injection process is highlycontrollable and tunable by changing the focal velocity and shaping the drivebeam current. Near-term experiments at FACET II where the capabilities togenerate tens of kA, <10 fs drivers are planned, could potentially producebeams with brightness near $10^{20}$ A/rad$^2$/m$^2$.

Ultra-Bright Electron Bunch Injection in a Plasma Wakefield Driven by a Superluminal Flying Focus Electron Beam