We consider ergodic backward stochastic differential equations (BSDEs), in a setting where noise is generated by a countable state uniformly ergodic Markov chain. We show that for Lipschitz drivers such that a comparison theorem holds, these equations admit unique solutions. To obtain this result, we show by coupling and splitting techniques that uniform ergodicity estimates of Markov chains are robust to perturbations of the rate matrix and that these perturbations correspond in a natural way to ergodic BSDEs. We then consider applications of this theory to Markov decision problems with a risk-averse average reward criterion.