The projected density profiles of gravitational lens galaxies are often assumed to be consistent with a simple elliptical power-law. Previous studies at low angular resolution have found this description to be satisfactory, but the limitations of this smooth model assumption have not yet been observationally tested. Here we report the analysis of a gravitational lens system observed at milli-arcsecond angular resolution with very long baseline interferometry at 1.7 GHz using a state-of-the-art Bayesian lens modelling technique for interferometric data. An elliptical power-law mass distribution is found to be a remarkably good description of the data, but there remain source-plane light components that are misaligned by several milli-arcseconds. The addition of angular multipole perturbations gives a significantly better fit (Bayes factor of +4751), while the best-focused source model is recovered using pixellated corrections to the lens potential. The projected density in our best model deviates from an elliptical power-law by 2.4 per cent (RMS) inside a region within 17 mas of the lensed images. The assumption of a perfect elliptical power-law would bias the measurement of H_0 by +0.6 per cent, and the observed flux ratios by up to 10.3 per cent, for this lens. Our results demonstrate that low-level density perturbations exist in this massive elliptical galaxy. Their presence can significantly affect studies of galaxy formation via flux-ratio anomalies, but do not strongly affect time-delay measurements of the Hubble constant.