We report on a near-infrared adaptive optics imaging survey of 31 young brown dwarfs and very low mass (VLM) stars, 28 of which are in the Chamaeleon I star-forming region, using the ESO Very Large Telescope. We resolve the suspected 0.16'' (~26 AU) binary Cha Hα 2 and present two new binaries, Hn 13 and CHXR 15, with separations of 0.13'' (~20 AU) and 0.30'' (~50 AU), respectively; the latter is one of the widest VLM systems known. We find a binary frequency of 11+9-6%, thus confirming the trend for a lower binary frequency with decreasing mass. By combining our work with previous surveys, we arrive at the largest sample of young VLM objects (72) with high angular resolution imaging to date. Its multiplicity fraction is in statistical agreement with that for VLM objects in the field. Furthermore, we note that many field stellar binaries with lower binding energies and/or wider cross sections have survived dynamical evolution and that statistical models suggest tidal disruption by passing stars is unlikely to affect the binary properties of our systems. Thus, we argue that there is no significant evolution of multiplicity with age among brown dwarfs and VLM stars in OB and T associations between a few megayears to several gigayears. Instead, the observations so far suggest that VLM objects are either less likely to be born in fragile multiple systems than solar-mass stars or such systems are disrupted very early. We dedicate this paper to the memory of our coauthor, Eduardo Delgado-Donate, who died in a hiking accident in Tenerife earlier this year.