Strong invariance principles describe the error term of a Brownian approximation to the partial sums of a stochastic process. While these strong approximation results have many applications, results for continuous-time settings have been limited. In this paper, we obtain strong invariance principles for a broad class of ergodic Markov processes. Strong invariance principles provide a unified framework for analysing commonly used estimators of the asymptotic variance in settings with a dependence structure. We demonstrate how this can be used to analyse the batch means method for simulation output of Piecewise Deterministic Monte Carlo samplers. We also derive a fluctuation result for additive functionals of ergodic diffusions using our strong approximation results.