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Automated systems for detecting deformation in satellite interferometric synthetic aperture radar (InSAR) imagery could be used to develop a global monitoring system for volcanic and urban environments. Here, we explore the limits of a convolutional neural networks for detecting slow, sustained deformations in wrapped interferograms. Using synthetic data, we estimate a detection threshold of 3.9 cm for deformation signals alone and 6.3 cm when atmospheric artifacts are considered. Overwrapping reduces this to 1.8 and 5.2 cm, respectively, as more fringes are generated without altering signal to noise ratio. We test the approach on time series of cumulative deformation from Campi Flegrei and Dallol, where overwrapping improves classification performance by up to 15%. We propose a mean‐filtering method for combining results of different wrap parameters to flag deformation. At Campi Flegrei, deformation of 8.5 cm/year was detected after 60 days and at Dallol, deformation of 3.5 cm/year was detected after 310 days. This corresponds to cumulative displacements of 3 and 4 cm consistent with estimates based on synthetic data. Key Points: We adapt a pretrained convolutional neural networks for detecting slow, steady deformation in InSAR imagesSynthetic tests show the detection threshold is 6.3 cm in volcanic environments but is reduced to 5.0 cm by decreasing wrapping intervalWe demonstrate the method on cumulative time series of interferograms from Campi Flegrei, Italy (8.5 cm/yearr) and Dallol, Ethopia (3.5 cm/year) [ABSTRACT FROM AUTHOR]