The observed electromagnetic emission from GW170817 suggests that a ‘cocoon’ of mildly relativistic material was released as a jet transferred its energy to the neutron-rich dynamical ejecta from the merger. On 17 August 2017, the detection of a gravitational-wave signature of merging neutron stars preceded that of a weak, short γ-ray burst. The source was quickly localized to the galaxy NGC 4993, and a range of telescopes measuring different parts of the electromagnetic spectrum followed the source, which was fading at most wavelengths. The radio emission, however, has continued to increase. Kunal Mooley and colleagues report continuing radio observations that are inconsistent with the off-axis relativistic jet that was initially believed to explain the optical, X-ray and γ-ray data. Instead, the radio data are best explained by a mildly relativistic wide-angle outflow. GW170817 was the first gravitational-wave detection of a binary neutron-star merger1. It was accompanied by radiation across the electromagnetic spectrum and localized2 to the galaxy NGC 4993 at a distance of 40 megaparsecs. It has been proposed that the observed γ-ray, X-ray and radio emission is due to an ultra-relativistic jet being launched during the merger (and successfully breaking out of the surrounding material), directed away from our line of sight (off-axis)3,4,5,6. The presence of such a jet is predicted from models that posit neutron-star mergers as the drivers of short hard-γ-ray bursts7,8. Here we report that the radio light curve of GW170817 has no direct signature of the afterglow of an off-axis jet. Although we cannot completely rule out the existence of a jet directed away from the line of sight, the observed γ-ray emission could not have originated from such a jet. Instead, the radio data require the existence of a mildly relativistic wide-angle outflow moving towards us. This outflow could be the high-velocity tail of the neutron-rich material that was ejected dynamically during the merger, or a cocoon of material that breaks out when a jet launched during the merger transfers its energy to the dynamical ejecta. Because the cocoon model explains the radio light curve of GW170817, as well as the γ-ray and X-ray emission (and possibly also the ultraviolet and optical emission)9,10,11,12,13,14,15, it is the model that is most consistent with the observational data. Cocoons may be a ubiquitous phenomenon produced in neutron-star mergers, giving rise to a hitherto unidentified population of radio, ultraviolet, X-ray and γ-ray transients in the local Universe.