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Extreme river flows can lead to inundation of floodplains, with consequent impacts for society, the environment, and the economy. Extreme flows are inherently difficult to model, being infrequent, irregularly spaced, and affected by nonstationary climatic controls. To identify patterns in extreme flows, a quantile regression approach can be used. This paper introduces a new framework for spatiotemporal quantile regression modeling, where the regression model is built as an additive model that includes smooth functions of time and space, as well as space-time interaction effects. The model exploits the flexibility that P-splines offer and can be easily extended to incorporate potential covariates. We propose to estimate model parameters using a penalized least squares regression approach as an alternative to linear programming methods, classically used in quantile parameter estimation. The model is illustrated on a data set of flows in 98 rivers across Scotland.