Mrk501 is a very high-energy (VHE) gamma-ray blazar located at z=0.034. During a period of two weeks in July 2014, the highest X-ray activity of Mrk501 was observed in ~14 years of operation of the Neil Gehrels Swift Gamma-ray Burst Observatory. We characterize the broadband variability of Mrk501 from radio to VHE gamma rays, and evaluate whether it can be interpreted within theoretical scenarios widely used to explain the broadband emission from blazars. The temporal evolution of the most prominent and variable segments of the SED is described with a one-zone synchrotron self-Compton model with variations in the break energy of the electron energy distribution (EED), and with some adjustments in the magnetic field strength and spectral shape of the EED. A narrow feature at ~3 TeV was observed in the VHE spectrum measured on 2014 July 19 (MJD 56857.98), which is the day with the highest X-ray flux (>0.3 keV) measured during the entire Swift mission. This feature is inconsistent with the classical analytic functions to describe the measured VHE spectra (power law, log-parabola, and log-parabola with exponential cutoff) at more than 3σ. A fit with a log-parabola plus a narrow component is preferred over the fit with a single log-parabola at more than 4σ, and a dedicated Monte Carlo simulation estimated the significance of this extra component to be larger than 3σ. Under the assumption that this VHE spectral feature is real, we show that it can be reproduced with three distinct theoretical scenarios: a) a pileup in the EED due to stochastic acceleration; b) a structured jet with two-SSC emitting regions, with one region dominated by an extremely narrow EED; and c) an emission from an IC pair cascade.