In plasmas with strongly anisotropic distribution functions $({T}_{\ensuremath{\parallel}b}/{T}_{\ensuremath{\perp}b}\ensuremath{\ll}1)$ 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 $({T}_{\ensuremath{\perp}b}\ensuremath{\gg}{T}_{\ensuremath{\parallel}b})$ to allow for nonaxisymmetric perturbations with $\ensuremath{\partial}/\ensuremath{\partial}\ensuremath{\theta}\ensuremath{\ne}0$. 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 ${s}_{b}={\stackrel{^}{\ensuremath{\omega}}}_{pb}^{2}/2{\ensuremath{\gamma}}_{b}^{2}{\ensuremath{\omega}}_{\ensuremath{\beta}\ensuremath{\perp}}^{2}\ensuremath{\gtrsim}0.5$ are linearly unstable to short-wavelength perturbations with ${k}_{z}^{2}{r}_{b}^{2}\ensuremath{\gtrsim}1$, provided the ratio of longitudinal and transverse temperatures is smaller than some threshold value. Here, ${\stackrel{^}{\ensuremath{\omega}}}_{pb}^{2}=4\ensuremath{\pi}{\stackrel{^}{n}}_{b}{e}_{b}^{2}/{\ensuremath{\gamma}}_{b}{m}_{b}$ is the relativistic plasma frequency squared, and ${\ensuremath{\omega}}_{\ensuremath{\beta}\ensuremath{\perp}}$ 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.