Even if local policymakers increasingly claim that tourism is one of the key factors of future economic development for the French small island La Réunion, international tourist arrivals are observed to be locked in a stagnation phase since the beginning of the 2000s. Starting from this stylized fact, this article aims to study if this phenomenon results from major external events hurting this economy regularly. Next, by using univariate unit root procedures with structural breaks, we test for evidence of permanent or transitory effects of external shocks on international tourist inflows (total, by source markets, and by category) over the period 1981–2015. Finally, the empirical analysis allows us to reject the null of a unit root. Then, stagnation of tourism arrivals to La Réunion is not due to exogenous shocks but probably results from endogenous impediments within the domestic tourism industry and unsuitable public policies.