Expectiles define a least squares analogue of quantiles. They have been the focus of a substantial quantity of research in the context of actuarial and financial risk assessment over the last 10 years. Unlike quantiles, expectiles induce coherent risk measures and are calculated using tail expectations rather than merely tail probabilities ; contrary to the popular quantile-based Expected Shortfall, they define elicitable risk measures. The behaviour and estimation of extreme expectiles using independent and identically distributed heavy-tailed observations has been investigated in a recent series of papers. The case of extreme conditional expectile estimation has, however, not been addressed so far in the literature. We build here a general theory for the estimation of extreme conditional expectiles in heteroscedastic regression models with heavy-tailed noise; an important feature of our approach is that it is intended to cope with covariates having a large but fixed dimension. We demonstrate how our results can be applied to a wide class of important examples, among which linear heteroscedastic models, heteroscedastic single-index models and autoregressive time series models. Our estimators are showcased on a numerical simulation study, as well as on real sets of actuarial and financial data.