A GridPix readout for a TPC based on the Timepix3 chip is developed for future applications at a linear collider. The GridPix detector consists of a gaseous drift volume read out by a single Timepix3 chip with an integrated amplification grid. Its performance is studied in a test beam with 2.5 GeV electrons. The GridPix detector detects single ionization electrons with high efficiency. The Timepix3 chip allowed for high sample rates and time walk corrections. Diffusion is found to be the dominating error on the track position measurement both in the pixel plane and in the drift direction, and systematic distortions in the pixel plane are below 10 μm. Using a truncated sum, an energy loss (dE/dx) resolution of 4.1% is found for an effective track length of 1 m.