We report on the first measurements of J/ψ production at very low transverse momentum (p_{T}<0.2  GeV/c) in hadronic Au+Au collisions at sqrt[s_{NN}]=200  GeV and U+U collisions at sqrt[s_{NN}]=193  GeV. Remarkably, the inferred nuclear modification factor of J/ψ at midrapidity in Au+Au (U+U) collisions reaches about 24 (52) for p_{T}<0.05  GeV/c in the 60%-80% collision centrality class. This noteworthy enhancement cannot be explained by hadronic production accompanied by cold and hot medium effects. In addition, the dN/dt distribution of J/ψ for the very low p_{T} range is presented for the first time. The distribution is consistent with that expected from the Au nucleus and shows a hint of interference. Comparison of the measurements to theoretical calculations of coherent production shows that the excess yield can be described reasonably well and reveals a partial disruption of coherent production in semicentral collisions, perhaps due to the violent hadronic interactions. Incorporating theoretical calculations, the results strongly suggest that the dramatic enhancement of J/ψ yield observed at extremely low p_{T} originates from coherent photon-nucleus interactions. In particular, coherently produced J/ψ's in violent hadronic collisions may provide a novel probe of the quark-gluon plasma.