Current interest in barocaloric effects has been stimulated by the discovery that these pressure-driven thermal changes can be giant near ferroic phase transitions in materials that display magnetic or electrical order. Here we demonstrate giant inverse barocaloric effects in the solid electrolyte AgI, near its superionic phase transition at ~420 K. Over a wide range of temperatures, hydrostatic pressure changes of 2.5 kbar yield large and reversible barocaloric effects, resulting in large values of refrigerant capacity. Moreover, the peak values of isothermal entropy change (60 J K-1 kg-1 or 0.34 J K-1 cm-3) and adiabatic temperature changes (18 K), which we identify for a starting temperature of 390 K, exceed all values previously recorded for barocaloric materials. Our work should therefore inspire the study of barocaloric effects in a wide range of solid electrolytes, as well as the parallel development of cooling devices.