Under conservative assumptions about the Galaxy, the derived velocity of the Smith Cloud indicates that it will have undergone at least one passage of the Galactic disc. Using hydrodynamic simulations, we examine the present-day structure of the Smith Cloud and find that a dark matter supported cloud is able to reproduce the observed present-day neutral hydrogen mass, column density distribution and morphology. In this case, the dark matter halo becomes elongated owing to the tidal interaction with the Galactic disc. Clouds in models neglecting dark matter confinement are destroyed upon disc passage, unless the initial cloud mass is well in excess of what is observed today. We then determine integrated flux upper limits to the gamma-ray emission around such a hypothesized dark matter core in the Smith Cloud. No statistically significant core or extended gamma-ray emission are detected down to a 95 per cent confidence level upper limit of 1.4 x 10(-10) ph cm(-2) s(-1) in the 1-300 GeV energy range. For the derived distance of 12.4 kpc, the Fermi upper limits set the first tentative constraints on the dark matter cross-sections annihilating into tau(+)tau(-) and b (b) over bar for a high-velocity cloud.