Temporally variant data observed on two-dimensional domains arise naturally across several disciplines. Functional data analysis proves to be inherently suitable for representing and modeling this kind of data, offering a rigorous mathematical framework capable of preserving the spatially continuous nature of these data. Within this framework, discrete-time evolving surfaces can be effectively modelled as functional time series of random real-valued functions defined on a two-dimensional domain. Building upon this approach, an anomaly detection method for handling such data is here developed. The proposal hinges on a probabilistic forecasting scheme for two-dimensional functional time series that incorporates conformal prediction bands for functional data. This methodology allows real-time construction of a prediction range for each point of the spatial domain ensuring joint control of the coverage probability. An anomaly is identified every time the observed surface deviates from the prediction bounds at a particular point in the domain. This approach inherently guarantees exact control over the probability of encountering one or more false warnings in the spatial domain, offering a viable solution for real-time monitoring of high-resolution spatial data. Finally, the proposed anomaly detection procedure is applied to a dataset collecting weekly interferometric measures of land elevation speed in the Phlegraean Fields volcanic area in Italy.