The concept of expectiles is a least squares analogue of quantiles. It has, over the last decade, received a fair amount of attention due to its potential for application in financial, actuarial, and economic contexts. Some very recent work has focused on the application of extreme expectiles to assess tail risk, and on their estimation in a heavy-tailed framework. We investigate the estimation of extreme conditional expectiles of a heavy-tailed random variable $Y$ given a finite-dimensional covariate $X$, whose dimension is allowed to be large. We derive generic conditions under which the limiting behaviour of our estimators can be investigated. We then present applications of our results to certain regression models of particular interest, as well as a finite-sample study to get a grasp of the behaviour of our procedures in practice.