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Magnetic fields grow quickly even at early cosmological times, suggesting the action of a small-scale dynamo (SSD) in the interstellar medium of galaxies. Many studies have focused on idealized turbulent driving of the SSD. Here we simulate more realistic supernova-driven turbulence to determine whether it can drive an SSD. Magnetic field growth occurring in our models appears inconsistent with simple tangling of magnetic fields, but consistent with SSD action, reproducing and confirming models by Balsara et al. (2004) that did not include physical resistivity $η$. We vary $η$, as well as the numerical resolution and supernova rate, $\dotσ$, to delineate the regime in which an SSD occurs. For a given $\dotσ$ we find convergence for SSD growth rate with resolution of a parsec. For $\dotσ\simeq\dotσ_{\rm sn}$, with $\dotσ_{\rm sn}$ the solar neighbourhood rate, the critical resistivity below which an SSD occurs is $0.005>η_{\rm crit}>0.001\,\rm kpc^{-1}\,\rm km s^{-1}$, and this increases with the supernova rate. Across the modelled range of 0.5--4 pc resolution we find that for $η<η_{\rm crit}$, the SSD saturates at about 5% of kinetic energy equipartition, independent of growth rate. In the range $0.2\dotσ_{\rm sn}\leq \dotσ\leq8\dotσ_{\rm sn}$ growth rate increases with $\dotσ$. SSDs in the supernova-driven interstellar medium commonly exhibit erratic growth.