  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 $N_\tau \rightarrow \infty$
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.
