This paper deals with risk-sensitive piecewise deterministic Markov decision processes, where the expected exponential utility of an infinite-horizon discounted cost is minimized. Both the transition rate and cost rate are allowed to be unbounded. Based on a dynamic programming observation, we introduce an auxiliary function with the time as an additional variable to analyze the problem, which is different from those with the risk-sensitive parameter as an additional variable in previous works. Under suitable assumptions, we derive the associated Feynman-Kac’s formula, and then establish the associated Hamilton–Jacobi–Bellman equation with the time as a differential variable, which leads to the existence of optimal policies depending on the time, explicitly showing that the risk-sensitive discounted optimal policies are not stationary.