Today's magnetic-field sensors are not capable of making measurements with both high spatial resolution and good field sensitivity. For example, magnetic force microscopy allows the investigation of magnetic structures with a spatial resolution in the nanometre range, but with low sensitivity, whereas SQUIDs and atomic magnetometers enable extremely sensitive magnetic-field measurements to be made, but at low resolution. Here we use one-dimensional Bose-Einstein condensates in a microscopic field-imaging technique that combines high spatial resolution (within 3 micrometres) with high field sensitivity (300 picotesla).