We present a search for f(J)(2220) production in radiative J/ψ→γf(J)(2220) decays using 460 fb⁻¹ of data collected with the BABAR detector at the SLAC PEP-II e(+)e⁻ collider. The f(J)(2220) is searched for in the decays to K(+)K⁻ and K(S)⁰K(S)⁰. No evidence of this resonance is observed, and 90% confidence level upper limits on the product of the branching fractions for J/ψ→γf(J)(2220) and f(J)(2220)→K(+)K⁻(K(S)⁰K(S)⁰) as a function of spin and helicity are set at the level of 10⁻⁵, below the central values reported by the Mark III experiment.