This paper describes Herschel observations of the nearby (8.5 pc) G5V multi-exoplanet host star 61 Vir at 70, 100, 160, 250, 350 and 500 m carried out as part of the DEBRIS survey. These observations reveal emission that is significantly extended out to a distance of >15 arcsec with a morphology that can be fitted by a nearly edge-on (77° inclination) radially broad (from 30 au out to at least 100 au) debris disc of fractional luminosity 2.7 × 10−5, with two additional (presumably unrelated) sources nearby that become more prominent at longer wavelengths. Chance alignment with a background object seen at 1.4 GHz provides potential for confusion, however, the star’s 1.4 arcsec yr−1 proper motion allows archival Spitzer 70 m images to confirm that what we are interpreting as disc emission really is circumstellar. Although the exact shape of the disc’s inner edge is not well constrained, the region inside 30 au must be significantly depleted in planetesimals. This is readily explained if there are additional planets outside those already known (i.e. in the 0.5–30 au region), but is also consistent with collisional erosion. We also find tentative evidence that the presence of detectable debris around nearby stars correlates with the presence of the lowest mass planets that are detectable in current radial velocity surveys. Out of an unbiased sample of the nearest 60 G stars, 11 are known to have planets, of which six (including 61 Vir) have planets that are all less massive than Saturn, and four of these have evidence for debris. The debris towards one of these planet hosts (HD 20794) is reported here for the first time. This fraction (4/6) is higher than that expected for nearby field stars (15 per cent), and implies that systems that form low-mass planets are also able to retain bright debris discs. We suggest that this correlation could arise because such planetary systems are dynamically stable and include regions that are populated with planetesimals in the formation process where the planetesimals can remain unperturbed over Gyr time-scales.