The existence of unconditional bases of reproducing kernels in the Fock-type spaces F φ with radial weights φ is studied. It is shown that there exist functions φ(r) of arbitrarily slow growth for which ln r = o(φ(r)) as r → ∞ and there are no unconditional bases of reproducing kernels in the space F φ . Thus, a criterion for the existence of unconditional bases cannot be given only in terms of the growth of the weight function.