The Large Hadron electron Collider (LHeC) is a proposed future particle-physics project colliding 60 GeV electrons from a six-pass recirculating energy-recovery linac (ERL) with 7 TeV protons stored in the LHC. The ERL technology allows for much higher beam current and, therefore, higher luminosity than a traditional linac. The high-current, high-energy electron beam can also be used to drive a free electron laser (FEL). In this study, we investigate the performance of an LHeC-based FEL, operated in the self-amplified spontaneous emission mode using electron beams after one or two turns, with beam energies of, e.g., 10, 20, 30 and 40 GeV, and aim at producing x-ray pulses at wavelengths ranging from 8 to 0.5 Å. In addition, we explore a possible path to use the 40 GeV electron beam for generating photon pulses at much lower wavelengths, down to a few picometer. We demonstrate that such ERL-based high-energy FEL would have the potential to provide orders of magnitude higher average brilliance at Å wavelengths than any other FEL either existing or proposed. It might also allow a pioneering step into the picometer wavelength regime.