- We present the results of a very deep (500 ks) Chandra observation, along with tailored numerical simulations, of the nearest, best resolved cluster cold front in the sky, which lies 90 kpc (19 arcmin) to the northwest of M 87. The northern part of the front appears the sharpest, with a width smaller than 2.5 kpc (1.5 Coulomb mean free paths; at 99 per cent confidence). Everywhere along the front, the temperature discontinuity is narrower than 4-8 kpc and the metallicity gradient is narrower than 6 kpc, indicating that diffusion, conduction and mixing are suppressed across the interface. Such transport processes can be naturally suppressed by magnetic fields aligned with the cold front. However, the northwestern part of the cold front is observed to have a nonzero width. The broadening is consistent with the presence of Kelvin-Helmholtz instabilities (KHI) on length scales of a few kpc. Based on comparison with simulations, the presence of KHI would imply that the effective viscosity of the intra-cluster medium is suppressed by more than an order of magnitude with respect to the isotropic Spitzer-like temperature dependent viscosity. Underneath the cold front, we observe quasi-linear features that are ~ 10 per cent brighter than the surrounding gas and are separated by ~ 15 kpc from each other in projection. Comparison to tailored numerical simulations suggests that the observed phenomena may be due to the amplification of magnetic fields by gas sloshing in wide layers below the cold front, where the magnetic pressure reaches ~ 5-10 per cent of the thermal pressure, reducing the gas density between the bright features.