We propose an extreme dimension reduction method extending the Extreme-PLS approach to the discretized functional framework, where the covariate lies in the infinite-dimensional Hilbert space $L^2([0,1])$ but is partially observed on a dense time grid. The ideas are partly borrowed from both Partial Least-Squares (PLS) and Sliced Inverse Regression (SIR) techniques. Accordingly, the method relies on the projection of the covariate onto a subspace and maximizes the covariance between its projection and the response conditionally on an extreme event capturing the tail-information. The covariate and the heavy-tailed response are supposed to be linked through a non-linear inverse single-index model and our goal is to infer the index in this regression framework. We propose a new family of estimators and show its asymptotic consistency with convergence rates under the model. Assuming mild conditions on the noise, most of the assumptions are stated in terms of regular variation unlike the standard literature on SIR and single-index regression. In addition, we expand the theoretical analysis with a model-free almost sure consistency result for the empirical tail-moments in a general separable Hilbert space. Finally, our results are illustrated on a finite-sample study with synthetic functional data as well as high-frequency financial data, highlighting the effectiveness of the dimension reduction for capturing tail dependence and for extreme risk management.