  We present a first search for dark-trident scattering in a neutrino beam
using a data set corresponding to $7.2 \times 10^{20}$ protons on target taken
with the MicroBooNE detector at Fermilab. Proton interactions in the neutrino
target at the Main Injector produce $\pi^0$ and $\eta$ mesons, which could
decay into dark-matter (DM) particles mediated via a dark photon $A^\prime$. A
convolutional neural network is trained to identify interactions of the DM
particles in the liquid-argon time projection chamber (LArTPC) exploiting its
image-like reconstruction capability. In the absence of a DM signal, we provide
limits at the $90\%$ confidence level on the squared kinematic mixing parameter
$\varepsilon^2$ as a function of the dark-photon mass in the range $10\le
M_{A^\prime}\le 400$ MeV. The limits cover previously unconstrained parameter
space for the production of fermion or scalar DM particles $\chi$ for two
benchmark models with mass ratios $M_{\chi}/M_{A^\prime}=0.6$ and $2$ and for
dark fine-structure constants $0.1\le\alpha_D\le 1$.
