Context. Wide-field surveys are a commonly-used method for studying thousands of objects simultaneously, to investigate, e.g., the joint evolution of star-forming galaxies and active galactic nuclei. Very long baseline interferometry (VLBI) observations can yield valuable input for such studies because they are able to identify AGN unambiguously in the moderate-to-high-redshift Universe. However, VLBI observations of large swaths of the sky are impractical using standard methods, because the fields of view of VLBI observations are on the order of 10 or less, and have therefore so far only played a minor role in galaxy evolution studies. Aims. We have embarked on a project to carry out Very Long Baseline Array (VLBA) observations of all 96 known radio sources in one of the best-studied areas in the sky, the Chandra deep field South (CDFS). The challenge was to develop methods that could significantly reduce the amount of observing (and post-processing) time, making such a project feasible. Methods. We developed an extension to the DiFX software correlator that allows one to efficiently correlate up to hundreds of positions within the primary beams of the interferometer antennas. This extension enabled us to target many sources simultaneously, at full resolution and high sensitivity, using only a small amount of observing time. The combination of wide fields-of-view and high sensitivity across the field in this survey is unprecedented. Results. We observed a single pointing containing the CDFS with the VLBA, in which 96 radio sources were known from previous observations with the Australia Telescope Compact Array (ATCA). From our input sample of 96 sources, 20 were detected with the VLBA, and one more source was tentatively detected. The majority of these objects have flux densities in agreement with arcsec-scale observations, implying that their radio emission comes from very small regions. Two objects are visibly resolved. One VLBI-detected object had been classified earlier as a star-forming galaxy. After comparing the VLBI detections to sources found in sensitive, colocated X-ray observations, we find that X-ray detections are not a good indicator for VLBI detections. Conclusions. We have successfully demonstrated a new extension to the DiFX software correlator, allowing one to observe hundreds of fields of view simultaneously. In a sensitive observation of the CDFS we detect 21% of the sources and were able to reclassify 7 sources as AGN, which had not been identified as such before. Wide-field VLBI survey science is now coming of age.