  A search for neutrino and antineutrino events correlated with 2,350 gamma-ray
bursts (GRBs) is performed with Borexino data collected between December 2007
and November 2015. No statistically significant excess over background is
observed. We look for electron antineutrinos ($\bar{\nu}_e$) that inverse beta
decay on protons with energies from 1.8\,MeV to 15\,MeV and set the best limit
on the neutrino fluence from GRBs below 8\,MeV. The signals from neutrinos and
antineutrinos from GRBs that scatter on electrons are also searched for, a
detection channel made possible by the particularly radio-pure scintillator of
Borexino. We obtain currently the best limits on the neutrino fluence of all
flavors and species below 7\,MeV. Finally, time correlations between GRBs and
bursts of events are investigated. Our analysis combines two semi-independent
data acquisition systems for the first time: the primary Borexino readout
optimized for solar neutrino physics up to a few MeV, and a fast waveform
digitizer system tuned for events above 1\,MeV.
