Expectiles define the only law-invariant, coherent and elicitable risk measure apart from the expectation. The popularity of expectile-based risk measures is steadily growing and their properties have been studied for independent data, but further results are needed to establish that extreme expectiles can be applied with the kind of dependent time series models relevant to finance. In this article we provide a basis for inference on extreme expectiles and expectile-based marginal expected shortfall in a general beta-mixing context that encompasses ARMA and GARCH models with heavy-tailed innovations. Our methods allow the estimation of marginal (pertaining to the stationary distribution) and dynamic (conditional on the past) extreme expectile-based risk measures. Simulations and applications to financial returns show that the new estimators and confidence intervals greatly improve on existing ones when the data are dependent.