A design of a compact free-electron laser (FEL), generating ultra-fast, high-peak flux, XUV pulses is presented. The FEL is driven by a high-current, 0.5 GeV electron beam from the Lawrence Berkeley National Laboratory (LBNL) laser-plasma accelerator, whose active acceleration length is only a few centimeters. The proposed ultra-fast source (~;;10 fs) would be intrinsically temporally synchronized to the drive laser pulse, enabling pump-probe studies in ultra-fast science. Owing to the high current (>10 kA) of the laser-plasma-accelerated electron beams, saturated output fluxes are potentially greater than 10^13 photons/pulse. Devices based both on self-amplified spontaneous emission and high-harmonic generated input seeds, to reduce undulator length and fluctuations, are considered.