Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ω) = h(t; ω) + ∫ k(t, τ; ω)f(τ, x(τ; ω)) dτ, whereω ∈ Ω, the supporting set of a probability measure space (Ω,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (∈, λ)-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].