We have carried out a multi-epoch spectroscopic survey of stars in the Arches cluster near the centre of our Galaxy, made in the K band with SINFONI and KMOS on the VLT. The Arches is one of the densest and most massive young open clusters in the Milky Way, containing around 100 O-type stars and a dozen Wolf-Rayet stars. By searching for significant radial velocity variability with amplitude exceeding 20 km/s, we have found evidence for likely binarity in a third of the bright targets studied, including Wolf-Rayets, O hypergiants and supergiants. Notably, three out of four highly X-ray-luminous WNL targets show significant radial velocity variability and are also detectable as bright radio sources, suggesting colliding wind binaries. One target is also photometrically variable, and has been modelled as a near-contact eclipsing SB2 binary with a 10.5 day orbital period, consisting of components of 82 and 60 solar masses: one of the most massive binaries dynamically measured in the Galaxy. Another, with a period of 13.4 days, appears strongly eccentric. These results run counter to earlier arguments that the most luminous Arches members are rejuvenated products of binary interaction and merger, implying a greater age for the Arches; instead, we find evidence for current pre-strong interaction binaries in the cluster, which supports a younger age around 2.5 Myr. Our findings also contribute to the understanding of the evolutionary pathways followed by the most massive stars.