ABSTRACT We report on a search for persistent radio emission from the one-off fast radio burst (FRB) 20190714A, as well as from two repeating FRBs, 20190711A and 20171019A, using the MeerKAT radio telescope. For FRB 20171019A, we also conducted simultaneous observations with the High-Energy Stereoscopic System (H.E.S.S.) in very high-energy gamma rays and searched for signals in the ultraviolet, optical, and X-ray bands. For this FRB, we obtain a UV flux upper limit of $1.39 \times 10^{-16}~{\rm erg\, cm^{-2}\, s^{-1}}$Å−1, X-ray limit of $∼ 6.6 \times 10^{-14}~{\rm erg\, cm^{-2}\, s^{-1}}$ and a limit on the very high energy gamma-ray flux $Φ (E\gt 120\, {\rm GeV}) \lt 1.7\times 10^{-12}\, \mathrm{erg\, cm^{-2}\, s^{-1}}$. We obtain a radio upper limit of ∼15 $μ$Jy beam−1 for persistent emission at the locations of both FRBs 20190711A and 20171019A with MeerKAT. However, we detected an almost unresolved (ratio of integrated flux to peak flux is ∼1.7 beam) radio emission, where the synthesized beam size was ∼ 8 arcsec size with a peak brightness of $∼ 53\, μ$Jy beam−1 at MeerKAT and $∼ 86\, μ$Jy beam−1 at e-MERLIN, possibly associated with FRB 20190714A at z = 0.2365. This represents the first detection of persistent continuum radio emission potentially associated with a (as-yet) non-repeating FRB. If the association is confirmed, one of the strongest remaining distinction between repeaters and non-repeaters would no longer be applicable. A parallel search for repeat bursts from these FRBs revealed no new detections down to a fluence of 0.08 Jy ms for a 1 ms duration burst.