Distant luminous Lyman-alpha emitters (LAEs) are excellent targets for spectroscopic observations of galaxies in the epoch of reionisation (EoR). We present deep high-resolution (R=5000) VLT/X-SHOOTER observations, along with an extensive collection of photometric data of `COLA1', a proposed double peaked LAE at z=6.6 (Hu et al. 2016). We rule out that COLA1's emission line is an [OII] doublet at z=1.475 on the basis of i) the asymmetric red line-profile and flux ratio of the peaks (blue/red=$0.31\pm0.03$) and ii) an unphysical [OII]/Ha ratio ([OII]/Ha > 22). We show that COLA1's observed B-band flux is explained by a faint extended foreground LAE, for which we detect Lya and [OIII] at z=2.142. We thus conclude that COLA1 is a real double-peaked LAE at z=6.593, the first discovered at z>6. COLA1 is UV luminous (M$_{1500}=-21.6\pm0.3$), has a high equivalent width (EW$_{0, Ly\alpha}$~85 \AA) and very compact Lya emission (r$_{50, Ly\alpha} = 0.3$ kpc). Relatively weak inferred Hb+[OIII] line-emission from Spitzer/IRAC indicates an extremely low metallicity of Z 6. Based on simple estimates, we find that COLA1 could have provided just enough photons to reionise its own ~0.3 pMpc (2.3 cMpc) bubble, allowing the blue Lya line to be observed. However, we also discuss alternative scenarios explaining the detected double peaked nature of COLA1. Our results show that future high-resolution observations of statistical samples of double peaked LAEs at z>5 are a promising probe of the occurrence of ionised regions around galaxies in the EoR.