We report on deep Chandra observations of the nearby broad-line radio galaxy Pictor A, which we combine with new Australia Telescope Compact Array (ATCA) observations. The new X-ray data have a factor 4 more exposure than observations previously presented and span a 15-year time baseline, allowing a detailed study of the spatial, temporal and spectral properties of the AGN, jet, hotspot and lobes. We present evidence for further time variation of the jet, though the flare that we reported in previous work remains the most significantly detected time-varying feature. We also confirm previous tentative evidence for a faint counterjet. Based on the radio through X-ray spectrum of the jet and its detailed spatial structure, and on the properties of the counterjet, we argue that inverse-Compton models can be conclusively rejected, and propose that the X-ray emission from the jet is synchrotron emission from particles accelerated in the boundary layer of a relativistic jet. For the first time, we find evidence that the bright western hotspot is also time-varying in X-rays, and we connect this to the small-scale structure in the hotspot seen in high-resolution radio observations. The new data allow us to confirm that the spectrum of the lobes is in good agreement with the predictions of an inverse-Compton model and we show that the data favour models in which the filaments seen in the radio images are predominantly the result of spatial variation of magnetic fields in the presence of a relatively uniform electron distribution.