The growing volume of synthetic aperture radar (SAR) imagery acquired by satellite constellations creates novel opportunities and opens new challenges for interferometric SAR (InSAR) applications to observe Earth’s surface processes and geohazards. In this paper, the Parallel Small BAseline Subset (P-SBAS) advanced InSAR processing chain running on the Geohazards Exploitation Platform (GEP) is trialed to process two unprecedentedly big stacks of Copernicus Sentinel-1 C-band SAR images acquired in 2014–2020 over a coastal study area in southern Italy, including 296 and 283 scenes in ascending and descending mode, respectively. Each stack was processed in the GEP in less than 3 days, from input SAR data retrieval via repositories, up to generation of the output P-SBAS datasets of coherent targets and their displacement histories. Use-cases of long-term monitoring of land subsidence at the Capo Colonna promontory (up −2.3 cm/year vertical and −1.0 cm/year east–west rate), slow-moving landslides and erosion landforms, and deformation at modern coastal protection infrastructure in the city of Crotone are used to: (i) showcase the type and precision of deformation products outputting from P-SBAS processing of big data, and the derivable key information to support value-adding and geological interpretation; and (ii) discuss potential and challenges of big data processing using cloud/grid infrastructure.