We present the results of our search for high-redshift Lyman-break galaxies over the GOODS-South field. We use Hubble Space Telescope (HST)-ACS data in B, V, i' & z', Very Large Telescope (VLT)-ISAAC J and Ks, Spitzer–Infrared Array Camera (IRAC) 3.6, 4.5, 5.8 and 8.0 μm data in conjunction with the new HAWK-I Y-band science verification data to search for dropout galaxies in the redshift range 6 < z < 9 . We survey ≈119 arcmin2 to YAB= 25.7 (5σ ), of which 37.5 arcmin2 reaches YAB= 25.9 . Candidate z' and Y dropouts were selected on the basis of a colour cut of (Y−J)AB > 0.75 mag and (z'−Y)AB > 1.0 mag, respectively. We find no robust Y-drops (z≈ 9) brighter than JAB < 25.4 . In our search for z'-band dropouts (z≈ 6.5– 7.5) , we identify four possible candidates, two with z'-drop colours and clear Spitzer–IRAC detections and two less likely candidates. We also identify two previously known Galactic T-dwarf stellar contaminants with these colours, and two likely transient objects seen in the Y-band data. The implications if all or none of our candidates is real on the ultraviolet galaxy luminosity functions at z > 6.5 are explored. We find our number of z'-drop candidates to be insufficient based on the expected number of z' drops in a simple no-evolution scenario from the z= 3 Lyman-break galaxy luminosity function but we are consistent with the observed luminosity function at z≈ 6 (if all our candidates are real). However, if one or both of our best z'-drop candidates are not z > 6.5 galaxies, this would demand evolution of the luminosity function at early epochs, in the sense that the number density of ultraviolet luminous star-forming galaxies at z > 7 is less than at z∼ 6 . We show that the future surveys to be conducted with the European Southern Observatory VISTA telescope over the next 5 yr will be able to measure the bulk of the luminosity function for both z' and Y dropouts and thus provide the strongest constraints on the level of star-formation within the epoch of reionization.