Transport properties, such as viscosity and thermal conduction, of the hot intergalactic plasma in clusters of galaxies are largely unknown. Whereas for laboratory plasmas these characteristics are derived from the gas density and temperature1, such recipes can be fundamentally different for the intergalactic plasma2 owing to a low rate of particle collisions and a weak magnetic field3. In numerical simulations, these unknowns can often be avoided by modelling these plasmas as hydrodynamic fluids4-6, even though local, non-hydrodynamic features observed in clusters contradict this assumption7-9. Using deep Chandra observations of the Coma Cluster10,11, we probe gas fluctuations in intergalactic medium down to spatial scales where the transport processes should prominently manifest themselves—provided that hydrodynamic models12 with pure Coulomb collision rates are indeed adequate. We do not find evidence of such transport processes, implying that the effective isotropic viscosity is orders of magnitude smaller than naively expected. This indicates either an enhanced collision rate in the plasma due to particle scattering off microfluctuations caused by plasma instabilities2,13,14 or that the transport processes are anisotropic with respect to the local magnetic field15. This also means that numerical models with high Reynolds number appear more consistent with observations. Our results demonstrate that observations of turbulence in clusters16,17 are giving rise to a branch of astrophysics that can sharpen theoretical views on galactic plasmas.