In this paper, we establish a version of the central limit theorem for Markov--Feller continuous time processes (with a Polish state space) that are exponentially ergodic in the bounded-Lipschitz distance and enjoy a continuous form of the Foster--Lyapunov condition. As an \hbox{example}, we verify the assumptions of our main result for a specific piecewise-deterministic Markov process, whose continuous component evolves according to semiflows, switched randomly at the jump times of a~Poisson process.