Detailed studies of relativistic jets in active galactic nuclei (AGN) require high-fidelity imaging at the highest possible resolution. This can be achieved using very long baseline interferometry (VLBI) at radio frequencies, combining worldwide (global) VLBI arrays of radio telescopes with a space-borne antenna on board a satellite. We present multiwavelength images made of the radio emission in the powerful quasar S5 0836+710, obtained using a global VLBI array and the antenna Spektr-R of the RadioAstron mission of the Russian Space Agency, with the goal of studying the internal structure and physics of the relativistic jet in this object. The RadioAstron observations at wavelengths of 18cm, 6cm, and 1.3cm are part of the Key Science Program for imaging radio emission in strong AGN. The internal structure of the jet is studied by analyzing transverse intensity profiles and modeling the structural patterns developing in the flow. The RadioAstron images reveal a wealth of structural detail in the jet of S5 0836+710 on angular scales ranging from 0.02mas to 200mas. Brightness temperatures in excess of 10^13^K are measured in the jet, requiring Doppler factors of >=100 for reconciling them with the inverse Compton limit. Several oscillatory patterns are identified in the ridge line of the jet and can be explained in terms of the Kelvin-Helmholtz (KH) instability. The oscillatory patterns are interpreted as the surface and body wavelengths of the helical mode of the KH instability. The interpretation provides estimates of the jet Mach number and of the ratio of the jet to the ambient density, which are found to be Mj~=12 and {eta}~=0.33. The ratio of the jet to the ambient density should be conservatively considered an upper limit because its estimate relies on approximations.