When a single‐species plasma is confined in a harmonic Penning trap at cryogenic temperature, the thermal equilibrium is approximately a uniform density spheroid (ellipsoid of revolution). Normal modes corresponding to quadrupole excitations of this plasma have recently been measured. In this paper, nonlinear equations of motion are derived for these quadrupole oscillations. For large amplitudes, the oscillations deform a spheroidal plasma into a triaxial ellipsoid with time‐dependent shape and orientation. The integrals of the motion are found and the cylindrically symmetric finite‐amplitude oscillations of a spheroid are studied. An analysis of all possible ellipsoidal equilibria is also carried out. New equilibria are discovered which correspond to finite‐amplitude versions of the noncylindrically symmetric linear quadrupole oscillations. The equilibria are shown to fall into two classes in which the ellipsoids are either tilted or aligned with respect to the magnetic field. Some of these equilibria have densities well above the Brillouin limit.