Ocean-driven basal melting of Antarctica’s floating ice shelves accounts for about half of their mass loss in steady state, where gains in ice-shelf mass are balanced by losses. Ice-shelf thickness changes driven by varying basal melt rates modulate mass loss from the grounded ice sheet and its contribution to sea level, and the changing meltwater fluxes influence climate processes in the Southern Ocean. Existing continent-wide melt-rate datasets have no temporal variability, introducing uncertainties in sea level and climate projections. Here, we combine surface height data from satellite radar altimeters with satellite-derived ice velocities and a new model of firn-layer evolution to generate a high-resolution map of time-averaged (2010–2018) basal melt rates and time series (1994–2018) of meltwater fluxes for most ice shelves. Total basal meltwater flux in 1994 (1,090 ± 150 Gt yr$^{–1}$) was similar to the steady-state value (1,100 ± 60 Gt yr$^{–1}$), but increased to 1,570 ± 140 Gt yr$^{–1}$ in 2009, followed by a decline to 1,160 ± 150 Gt yr$^{–1}$ in 2018. For the four largest ‘cold-water’ ice shelves, we partition meltwater fluxes into deep and shallow sources to reveal distinct signatures of temporal variability, providing insights into climate forcing of basal melting and the impact of this melting on the Southern Ocean.