We study a non-resonant instability of kinetic Alfvén waves (KAWs) driven by compensated currents. Such currents set up in response to energetic ion beams occurring in many space and astrophysical plasmas, like foreshock regions in the solar wind and around supernova remnants. Kinetic effects of the background ion gyroradius make the KAW instability stronger than its magnetohydrodynamic (MHD) counterpart and shift its maximum to shorter wavelengths. The KAW growth time can be very short, approaching the proton gyroperiod in the terrestrial foreshock ahead of the quasi-perpendicular bow shock region. The oblique Alfvén instability driven by the cosmic rays in the interstellar and intergalactic plasmas develops mostly in the MHD regime and can extend in the KAW regime only at large fluxes of cosmic rays. Short cross-field wavelengths of growing Alfvén modes facilitate stochastic cross-field acceleration of cosmic rays.