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The NeuLAND (New Large-Area Neutron Detector) plastic-scintillator-based time-of-flight detector for 0.1-1.6 GeV neutrons is currently under construction at the Facility for Antiproton and Ion Research (FAIR), Darmstadt, Germany. In its final configuration, NeuLAND will consist of 3000 2.7 m $\times$ 5 cm $\times$ 5 cm big plastic scintillator bars that are read out on each end by fast timing photomultipliers. Here, data from a comprehensive study of an alternative light readout scheme using silicon photomultipliers (SiPM) are reported. For this purpose, a NeuLAND bar was instrumented on each end with a SiPM-based prototype of the same geometry as a 1'' photomultiplier tube, including four 6 $\times$ 6 mm$^2$ SiPMs, amplifiers, high voltage supply, and microcontroller. Tests were carried out using the 35 MeV electron beam from the superconducting Electron Linac for beams with high Brilliance and low Emittance (ELBE) with its picosecond-level time jitter in two different modes of operation, namely parasitic mode with one electron per bunch and single-user mode with 1-60 electrons per bunch. Acqiris fast digitisers were used for data acquisition. In addition, off-beam tests using cosmic rays and the NeuLAND data acquisition scheme have been carried out. Typical time resolutions of $σ_t\leq$ 120 ps were found for $\geq$95% efficiency for minimum ionising particles, improving on previous work at ELBE and exceeding the NeuLAND timing goal of $σ_t$ < 150 ps. Over a range of 10-300 MeV deposited energy in the NeuLAND bar, the gain was found to deviate by $\leq$10% ($\leq$20%) from linearity for 35 mm (75 mm) SiPM pitch, respectively, satisfactory for calorimetric use of the full NeuLAND detector. The dark rate of the prototype studied was found to be lower than the expected cosmic-ray induced background in NeuLAND.