Quantiles are recognized tools for risk management and can be seen as minimizers of an L1-loss function, but do not define coherent risk measures in general. Expectiles, meanwhile, are minimizers of an L2-loss function and define coherent risk measures; they have started to be considered as good alternatives to quantiles in insurance and finance. Quantiles and expectiles belong to the wider family of Lp-quantiles. We propose here to construct kernel estimators of extreme conditional Lp-quantiles. We study their asymptotic properties in the context of conditional heavy-tailed distributions and we show through a simulation study that taking p ∈ (1, 2) may allow to recover extreme conditional quantiles and expectiles accurately. Our estimators are also showcased on a real insurance data set.