We continue our study of heavy-light four-quark states and find evidence from lattice QCD for the existence of a strong-interaction-stable $I\left(J^P\right)=0\left(1^{+}\right)$ $ud\overline{c}\overline{b}$ tetraquark with mass in the range of 15 to 61 MeV below $\overline{D}{B}^{*}$ threshold. Since this range includes the electromagnetic $\overline{D}B\gamma$ decay threshold, current uncertainties do not allow us to determine whether such a state would decay electromagnetically or only weakly. We also perform a study at fixed pion mass, with non-relativistic QCD (NRQCD) for the heavy quarks, simulating $q{q}^{\prime }{\overline{b}}^{\prime }\overline{b}$ and $q{q}^{\prime }{\overline{b}}^{\prime }{\overline{b}}^{\prime }$ tetraquarks with $q$, $q^{\prime }=ud$ or $\ell s$ and variable, unphysical $m_{b^{\prime }}$ in order to investigate the heavy mass dependence of such tetraquark states. We find that the dependence of the binding energy follows a phenomenologically expected form and that, though NRQCD breaks down before $m_{b^{\prime }}=m_c$ is reached, the results at higher $m_{b^{\prime }}$ clearly identify the $ud{\overline{b}}^{\prime }\overline{b}$ channel as the most likely to support a strong-interaction-stable tetraquark state at $m_{b^{\prime }}=m_c$. This observation serves to motivate the direct $ud\overline{c}\overline{b}$ simulation. Throughout we use dynamical $n_f=2+1$ ensembles with pion masses $m_{\pi }=415$, 299, and 164 MeV reaching down almost to the physical point, a relativistic heavy quark prescription for the charm quark and NRQCD for the bottom quark(s).