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We present results from our analysis of the Hydra I cluster observed in neutral atomic hydrogen (H I) as part of the Widefield ASKAP L-band Legacy All-sky Blind Survey (WALLABY). These WALLABY observations cover a 60-square-degree field of view with uniform sensitivity and a spatial resolution of 30 arcsec. We use these wide-field observations to investigate the effect of galaxy environment on H I gas removal and star formation quenching by comparing the properties of cluster, infall, and field galaxies extending up to ∼5R200 from the cluster centre. We find a sharp decrease in the H I-detected fraction of infalling galaxies at a projected distance of ∼1.5R200 from the cluster centre from ∼85 per cent to ∼35 per cent. We see evidence for the environment removing gas from the outskirts of H I-detected cluster and infall galaxies through the decrease in the H I to r-band optical disc diameter ratio. These galaxies lie on the star-forming main sequence, indicating that gas removal is not yet affecting the inner star-forming discs and is limited to the galaxy outskirts. Although we do not detect galaxies undergoing galaxy-wide quenching, we do observe a reduction in recent star formation in the outer disc of cluster galaxies, which is likely due to the smaller gas reservoirs present beyond the optical radius in these galaxies. Stacking of H I non-detections with H I masses below MHI≲108.4M⊙ will be required to probe the H I of galaxies undergoing quenching at distances ≳60 Mpc with WALLABY. © 2021 The Author(s). ; This research was conducted by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. The Australian SKA Pathfinder is part of the Australia Telescope National Facility which is managed by the Commonwealth Scientific and Industrial Research Organisation (CSIRO). Operation of ASKAP is funded by the Australian Government with support from the National Collaborative Research Infrastructure Strategy. ASKAP uses the resources of the Pawsey Supercomputing Centre. Establishment of ASKAP, the Murchison Radio-astronomy Observatory (MRO), and the Pawsey Supercomputing Centre are initiatives of the Australian Government, with support from the Government of Western Australia and the Science and Industry Endowment Fund. We acknowledge the Wajarri Yamatji as the traditional owners of the Observatory site. We also thank the MRO site staff. This paper includes archived data obtained through the CSIRO ASKAP Science Data Archive, CASDA (http://data.csiro.au). LC acknowledges support from the Australian Research Council Discovery Project and Future Fellowship funding schemes (DP210100337, FT180100066). DO is a recipient of an Australian Research Council Future Fellowship (FT190100083) funded by the Australian Government. JR acknowledges support from the State Research Agency (AEI-MCINN) of the Spanish Ministry of Science and Innovation under the grant 'The structure and evolution of galaxies and their central regions' with reference PID2019-105602GB-I00/10.13039/501100011033. LVM acknowledges financial support from the State Agency for Research of the Spanish Ministry of Science, Innovation and Universities through the 'Center of Excellence Severo Ochoa' awarded to the Instituto de Astrofísica de Andalucía (SEV-2017-0709), from the grant RTI2018-096228-B-C31 (Ministry of Science, Innovation and Universities/State Agency for Research/European Regional Development Funds, European Union), from the grant RED2018-102587-T (Spanish Ministry of Science, Innovation and Universities/State Agency for Research), and grant IAA4SKA (Ref. P18-RT-3082) from the Consejeria de Transformacion Economica, Industria, Conocimiento y Universidades de la Junta de Andalucia and the European Regional Development Fund from the European Union. AB acknowledges support from the Centre National d'Etudes Spatiales (CNES), France. ARHS acknowledges receipt of the Jim Buckee Fellowship at ICRAR-UWA. SHO acknowledges support from the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT: MSIT) (No. NRF-2020R1A2C1008706). This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 679627; project name FORNAX). This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. This publication makes use of data products from the Wide-field Infrared Survey Explorer (WISE), which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This work is based in part on observations made with the Galaxy Evolution Explorer (GALEX). GALEX is a NASA Small Explorer, whose mission was developed in cooperation with the Centre National d'Etudes Spatiales (CNES) of France and the Korean Ministry of Science and Technology. GALEX is operated for NASA by the California Institute of Technology under NASA contract NAS5-98034. ; Peer reviewed