We present a first search for dark-trident scattering in a neutrino beam using a dataset corresponding to 7.2×10^{20} protons on target taken with the MicroBooNE detector at Fermilab. Proton interactions in the neutrino target at the main injector produce π^{0} and η mesons, which could decay into dark-matter (DM) particles mediated via a dark photon A^{'}. A convolutional neural network is trained to identify interactions of the DM particles in the liquid-argon time projection chamber (LArTPC) exploiting its imagelike reconstruction capability. In the absence of a DM signal, we provide limits at the 90% confidence level on the squared kinematic mixing parameter ϵ^{2} as a function of the dark-photon mass in the range 10≤M_{A^{'}}≤400 MeV. The limits cover previously unconstrained parameter space for the production of fermion or scalar DM particles χ for two benchmark models with mass ratios M_{χ}/M_{A^{'}}=0.6 and 2 and for dark fine-structure constants 0.1≤α_{D}≤1.