The blazar Mrk 501 (z=0.034) was observed at very-high-energy (VHE, $Egtrsim 100$~GeV) gamma-ray wavelengths during a bright flare on the night of 2014 June 23-24 (MJD 56832) with the H.E.S.S. phase-II array of Cherenkov telescopes. Data taken that night by H.E.S.S. at large zenith angle reveal an exceptional number of gamma-ray photons at multi-TeV energies, with rapid flux variability and an energy coverage extending significantly up to 20 TeV. This data set is used to constrain Lorentz invariance violation (LIV) using two independent channels: a temporal approach considers the possibility of an energy dependence in the arrival time of gamma rays, whereas a spectral approach considers the possibility of modifications to the interaction of VHE gamma rays with extragalactic background light (EBL) photons. The non-detection of energy-dependent time delays and the non-observation of deviations between the measured spectrum and that of a supposed power-law intrinsic spectrum with standard EBL attenuation are used independently to derive strong constraints on the energy scale of LIV ($E_{rm{QG}}$) in the subluminal scenario for linear and quadratic perturbations in the dispersion relation of photons. For the case of linear perturbations, the 95% confidence level limits obtained are $E_{rm{QG},1} > 3.6 times 10^{17} rm{GeV} $ using the temporal approach and $E_{rm{QG},1} > 2.6 times 10^{19} rm{GeV}$ using the spectral approach. For the case of quadratic perturbations, the limits obtained are $E_{rm{QG},2} > 8.5 times 10^{10} rm{GeV} $ using the temporal approach and $E_{rm{QG},2} > 7.8 times 10^{11} rm{ GeV}$ using the spectral approach.
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