It has been speculated that Lorentz-invariance violation (LIV) might be generated by quantum-gravity (QG) effects. As a consequence, particles may not travel at the universal speed of light. In particular, superluminal extragalactic neutrinos would rapidly lose energy via the bremssthralung of electron-positron pairs (nu -> nu e(+)e(-)), damping their initial energy into electromagnetic cascades, a figure constrained by Fermi-Large Area Telescope data. We show that the two cascade neutrino events with energies around 1 PeV recently detected by IceCube-if attributed to extragalactic diffuse events, as it appears likely-can place the strongest bound on LIV in the neutrino sector, namely, delta = (v(2) - 1) < O(10(-18)), corresponding to a QG scale M-QG greater than or similar to 10(5)M(Pl) (M-QG greater than or similar to 10(-4)M(Pl)) for a linear (quadratic) LIV, at least for models inducing superluminal neutrino effects (delta > 0).