The vast, deep, volatile-ice-filled basin informally named Sputnik Planum is central to Pluto's vigorous geological activity. Composed of molecular nitrogen, methane, and carbon monoxide ices, but dominated by nitrogen ice, this layer is organized into cells or polygons, typically about 10 to 40 kilometres across, that resemble the surface manifestation of solid-state convection. Here we report, on the basis of available rheological measurements, that solid layers of nitrogen ice with a thickness in excess of about one kilometre should undergo convection for estimated present-day heat-flow conditions on Pluto. More importantly, we show numerically that convective overturn in a several-kilometre-thick layer of solid nitrogen can explain the great lateral width of the cells. The temperature dependence of nitrogen-ice viscosity implies that the ice layer convects in the so-called sluggish lid regime, a unique convective mode not previously definitively observed in the Solar System. Average surface horizontal velocities of a few centimetres a year imply surface transport or renewal times of about 500,000 years, well under the ten-million-year upper-limit crater retention age for Sputnik Planum. Similar convective surface renewal may also occur on other dwarf planets in the Kuiper belt, which may help to explain the high albedos shown by some of these bodies.