We present the first results from the KMOS AGN Survey at High redshift (KASHz), a VLT/KMOS integral-field spectroscopic survey of z>0.6 AGN. We present galaxy-integrated spectra of 89 X-ray AGN (Lx=10^42-10^45 erg/s), for which we observed [O III] (z=1.1-1.7) or Halpha emission (z=0.6-1.1). The targets have X-ray luminosities representative of the parent AGN population and we explore the emission-line luminosities as a function of X-ray luminosity. For the [O III] targets, ~50 per cent have ionised gas velocities indicative of gas that is dominated by outflows and/or highly turbulent material (i.e., overall line-widths >~600 km/s). The most luminous half (i.e., Lx>6x10^43 erg/s) have a >~2 times higher incidence of such velocities. On the basis of our results, we find no evidence that X-ray obscured AGN are more likely to host extreme kinematics than unobscured AGN. Our KASHz sample has a distribution of gas velocities that is consistent with a luminosity-matched sample of z<0.4 AGN. This implies little evolution in the prevalence of ionised outflows, for a fixed AGN luminosity, despite an order-of-magnitude decrease in average star-formation rates over this redshift range. Furthermore, we compare our Halpha targets to a redshift-matched sample of star-forming galaxies and despite a similar distribution of Halpha luminosities and likely star-formation rates, we find extreme ionised gas velocities are up to ~10x more prevalent in the AGN-host galaxies. Our results reveal a high prevalence of extreme ionised gas velocities in high-luminosity X-ray AGN and imply that the most powerful ionised outflows in high-redshift galaxies are driven by AGN activity.