This paper deals with the optimal stopping problem for a class of strong, nonstandard Markov processes the paths of which follow continuous deterministic trajectories but for random jumps at random times. The gain function, defined on the state space, is assumed to be bounded, measurable and, roughly speaking, continuous along the trajectories of the deterministic drift Combining continuous-time deterministic maximization and discrete-time dynamic programming yields a functional operator solving the single-jump problem.