A 3‐D theory is presented for a free electron laser that employs an electron beam of a thickness comparable to both the wiggler wavelength and the waveguide radius. The time‐independent and the linearized time‐dependent cold fluid and Maxwell equations are expanded in a small parameter, which is the ratio of the perpendicular to parallel electron momentum. The stability problem is reduced to a nonlinear eigenvalue problem of a fourth‐order system of linear ordinary differential equations. A perturbation method is justified and used to solve these equations. A dispersion relation is derived which results from the solvability condition for the first‐order equations in the perturbation. The orders of magnitude of the beam density and wave frequency, for which the growth rate of the instability scales as in the strong‐pump regime of the 1‐D analysis, are determined. An equation, which the beam energy radial profile has to satisfy, is also derived.