<div>This work has been motivated by the challenge of the 2017 conference on Extreme-Value Analysis (EVA2017), with the goal of predicting daily precipitation quantiles at the $99.8\%$ level for each month at observed and unobserved locations. To propose a general approach to this specific problem, we develop a Bayesian generalized additive modeling framework tailored to estimate complex trends in marginal extremes observed at a large number of spatio-temporal locations. Our approach is based on a set of regression equations linked to the exceedance probability above a high threshold and to the size of the excess, the latter being modeled using the asymptotic generalized Pareto (GP) distribution suggested by Extreme-Value Theory. Latent random effects are modeled additively and semi-parametrically using Gaussian process priors, which provides high flexibility and interpretability. Fast and accurate estimation of posterior distributions is performed thanks to the Integrated Nested Laplace approximation (INLA), efficiently implemented in the R-INLA software. We show that the GP distribution meets the theoretical requirements of INLA, and we then develop a penalized complexity prior specification for the tail index, which is a crucial parameter for extrapolating tail event probabilities. This prior concentrates mass close to a light exponential tail while still allowing heavier tails by penalizing the distance to the exponential distribution at a constant rate. We illustrate this new methodological framework through the modeling of spatial and seasonal trends in daily precipitation data provided by the EVA2017 challenge. Capitalizing on R-INLA's fast computation capacities and powerful distributed computing resources, we conduct an extensive cross-validation study to select model parameters governing the smoothness of trends, which are critical for accurate prediction. Our results clearly outperform simple benchmarks and are comparable to the best-scoring approach among the other teams.</div>