Understanding the spatial extent of extreme precipitation is necessary for determining flood risk and adequately designing infrastructure (e.g., stormwater pipes) to withstand such hazards. While environmental phenomena typically exhibit weakening spatial dependence at increasingly extreme levels, limiting max-stable process models for block maxima have a rigid dependence structure that does not capture this type of behavior. We propose a flexible Bayesian model from a broader family of (conditionally) max-infinitely divisible processes that allows for weakening spatial dependence at increasingly extreme levels, and due to a hierarchical representation of the likelihood in terms of random effects, our inference approach scales to large datasets. Therefore, our model not only has a flexible dependence structure, but it also allows for fast, fully Bayesian inference, prediction and conditional simulation in high dimensions. The proposed model is constructed using flexible random basis functions that are estimated from the data, allowing for straightforward inspection of the predominant spatial patterns of extremes. In addition, the described process possesses (conditional) max-stability as a special case, making inference on the tail dependence class possible. We apply our model to extreme precipitation in North-Eastern America, and show that the proposed model adequately captures the extremal behavior of the data. Interestingly, we find that the principal modes of spatial variation estimated from our model resemble observed patterns in extreme precipitation events occurring along the coast (e.g., with localized tropical cyclones and convective storms) and mountain range borders. Our model, which can easily be adapted to other types of environmental datasets, is therefore useful to identify extreme weather patterns and regions at risk. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.