  The blazar Mrk 501 (z=0.034) was observed at very-high-energy (VHE, $E\gtrsim
100$~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.
