We present results for lattice QCD with staggered fermions in the limit of infinite gauge coupling, obtained from a worm-type Monte Carlo algorithm on a discrete spatial lattice but with continuous Euclidean time. This is obtained by sending both the anisotropy parameter $\xi =a_{\sigma }/a_{\tau }$ and the number of time slices $N_{\tau }$ to infinity, keeping the ratio $aT=\xi /N_{\tau }$ fixed. The obvious gain is that no continuum extrapolation $N_{\tau }\to \infty$ has to be carried out. Moreover, the algorithm is faster, and the sign problem disappears. We derive the continuous time partition function and the corresponding Hamiltonian formulation. We compare our computations with those on discrete lattices and study both zero and finite temperature properties of lattice QCD in this regime.