Consider a topologically transitive unilateral countable Markov shift $Σ$, a locally constant potential $φ: Σ\to \mathbb{R}$ satisfying suitable conditions, and assume that $μ_t$ is the unique stationary Markov equilibrium state associated to the potential $tφ$ for each $t \geq 1$. In this paper we prove a first level large deviations principle at zero temperature for the family of equilibrium states $(μ_t)_{t \geq 1}$ and we extend the result to the setting of bilateral countable Markov shifts.