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This paper analyzes the existence of Nash equilibrium in a discrete-time Markov stopping game with two players. At each decision point, Player II is faced with the choice of either ending the game and thus granting Player I a final reward or letting the game continue. In the latter case, Player I performs an action that affects transitions and receives a running reward from Player II. We assume that Player I has a constant and non-zero risk sensitivity coefficient, while Player II strives to minimize the utility of Player I. The effectiveness of decision strategies was measured by the risk-sensitive expected total reward of Player I. Exploiting mild continuity-compactness conditions and communication-ergodicity properties, we found that the value function of the game is described as a single fixed point of the equilibrium operator, determining a Nash equilibrium. In addition, we provide an illustrative example in which our assumptions hold. [ABSTRACT FROM AUTHOR]