  This paper presents the results of 475 hours of interferometric observations
with the Australia Telescope Compact Array towards the Spiderweb protocluster
at \(z=2.16\). We search for large, extended molecular gas reservoirs among 46
previously detected CO(1-0) emitters, employing a customised method we
developed. Based on the CO emission images and position-velocity diagrams, as
well as the ranking of sources using a binary weighting of six different
criteria, we have identified 14 robust and 7 tentative candidates that exhibit
large extended molecular gas reservoirs. These extended reservoirs are defined
as having sizes greater than 40 kpc or super-galactic scale. This result
suggests a high frequency of extended gas reservoirs, comprising at least \(30
\%\) of our CO-selected sample. An environmental study of the candidates is
carried out based on N-th nearest neighbour and we find that the large
molecular gas reservoirs tend to exist in denser regions. The spatial
distribution of our candidates is mainly centred on the core region of the
Spiderweb protocluster. The performance and adaptability of our method are
discussed. We found 13 (potentially) extended gas reservoirs located in nine
galaxy (proto)clusters from the literature. We noticed that large extended
molecular gas reservoirs surrounding (normal) star-forming galaxies in
protoclusters are rare. This may be attributable to the lack of observations
low-J CO transitions and the lack of quantitative analyses of molecular gas
morphologies. The large gas reservoirs in the Spiderweb protocluster are a
potential source of the intracluster medium seen in low redshift Virgo- or
Coma-like galaxy clusters.
