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In plasmas with strongly anisotropic distribution functions (T_{∥b}/T_{⊥b}≪1) a Harris-like collective instability may develop if there is sufficient coupling between the transverse and longitudinal degrees of freedom. Such anisotropies develop naturally in accelerators and may lead to a deterioration of beam quality. This paper extends previous numerical studies [E. A. Startsev, R. C. Davidson, and H. Qin, Phys. Plasmas 9, 3138 (2002)] of the stability properties of intense non-neutral charged particle beams with large temperature anisotropy (T_{⊥b}≫T_{∥b}) to allow for nonaxisymmetric perturbations with ∂/∂θ≠0. The most unstable modes are identified, and their eigenfrequencies, radial mode structure, and nonlinear dynamics are determined. The simulation results clearly show that moderately intense beams with s_{b}=ω[over ^]_{pb}^{2}/2γ_{b}^{2}ω_{β⊥}^{2}≳0.5 are linearly unstable to short-wavelength perturbations with k_{z}^{2}r_{b}^{2}≳1, provided the ratio of longitudinal and transverse temperatures is smaller than some threshold value. Here, ω[over ^]_{pb}^{2}=4πn[over ^]_{b}e_{b}^{2}/γ_{b}m_{b} is the relativistic plasma frequency squared, and ω_{β⊥} is the betatron frequency associated with the applied smooth-focusing field. A theoretical model is developed based on the Vlasov-Maxwell equations which describes the essential features of the linear stages of instability. Both the simulations and the analytical theory predict that the dipole mode (azimuthal mode number m=1) is the most unstable mode. In the nonlinear stage, tails develop in the longitudinal momentum distribution function, and the kinetic instability saturates due to resonant wave-particle interactions.