We report the detection of [CII]158um emission from a system of three closely-separated sources in the COSMOS field at z~4.56, as part of the ALMA Large Program to INvestigate CII at Early times (ALPINE). The two dominant sources are closely associated, both spatially (1.6"~11kpc) and in velocity (~100km/s), while the third source is slightly more distant (2.8"~18kpc, ~300km/s). The second strongest source features a slight velocity gradient, while no significant velocity gradient is seen in the other two sources. Using the observed [CII] luminosities, we derive a total log(SFR_[CII]/[Msol/year])=2.8+/-0.2, which may be split into contributions of 59%, 31%, and 10% from the central, east, and west sources, respectively. Comparison of these [CII] detections to recent zoom-in cosmological simulations suggests an ongoing major merger. We are thus witnessing a system in a major phase of mass build-up by merging, including an on-going major merger and an upcoming minor merger, which is expected to end up in a single massive galaxy by z~2.5.