The extension of gyrokinetic theory to arbitrary frequencies by Chen and Tsai [Phys. Fluids26, 141 (1983); Plasma Phys.25, 349 (1983)] is used to study cyclotron absorption in a straight magnetic field with a perpendicular, linear gradient in strength. The analysis includes the effects of magnetic field variation across the Larmor orbit and is restricted to propagation perpendicular to the field. It yields the following results for propagation into the field gradient. The standard optical depths for the fundamental O‐mode and second harmonic X‐mode resonances are obtained from the absorption profiles given in this paper, without invoking relativistic mass variation [see also Antonsen and Manheimer, Phys. Fluids21, 2295 (1978)]. The compressional Alfve´n wave is shown to undergo perpendicular cyclotron damping at the fundamental minority resonance in a two‐ion species plasma and at second harmonic resonance in a single‐ion species plasma. Ion Bernstein waves propagating into the second harmonic resonance are no longer unattenuated, but are increasingly damped as they approach the resonance. It is shown how the kinetic power flow affects absorption profiles, yielding information previously obtainable only from full‐wave theory. In all cases, the perpendicular cyclotron damping arises from the inclusion of magnetic field variation across the Larmor orbit.