We have carried out a sensitive high-resolution imaging survey of stars in the young (6-8 Myr), nearby (97 pc) compact cluster around η Chamaeleontis to search for stellar and substellar companions. Our data were obtained using the NACO adaptive optics system on the ESO Very Large Telescope (VLT). Given its youth and proximity, any substellar companions are expected to be luminous, especially in the near-infrared, and thus easier to detect next to their parent stars. Here, we present VLT NACO adaptive optics imaging with companion detection limits for 17 η Cha cluster members, and follow-up VLT ISAAC near-infrared spectroscopy for companion candidates. The widest binary detected is ~02, corresponding to the projected separation 20 AU, despite our survey being sensitive down to substellar companions outside 03, and planetary-mass objects outside 05. This implies that the stellar companion probability outside 03 and the brown dwarf companion probability outside 05 are less than 0.16 with 95% confidence. We compare the wide binary frequency of η Cha to that of the similarly aged TW Hydrae association and estimate the statistical likelihood that the wide binary probability is equal in both groups to be less than 2 × 10-4. Even though the η Cha cluster is relatively dense, stellar encounters in its present configuration cannot account for the relative deficit of wide binaries. We thus conclude that the difference in wide binary probability in these two groups provides strong evidence for multiplicity properties being dependent on environment. In two appendices we derive the projected separation probability distribution for binaries, used to constrain physical separations from observed projected separations, and summarize statistical tools useful for multiplicity studies.