A fully nonlinear theory for stationary whistler waves propagating parallel to the ambient magnetic field in a cold plasma has been developed. It is shown that in the wave frame proton dynamics must be included in a self-consistent manner. The complete system of nonlinear equations can be reduced to two coupled differential equations for the transverse electron or proton speed and its phase, and these possess a phase-portrait integral which provides the main features of the dynamics of the system. Exact analytical solutions are found in the approximation of ‘small’ (but nonlinear) amplitudes. A soliton-type solution with a core filled by smaller-scale oscillations (called ‘oscillitons’) is found. The dependence of the soliton amplitude on the Alfvén Mach number, and the critical soliton strength above which smooth soliton solutions cannot be constructed is also found. Another interesting class of solutions consisting of a sequence of wave packets exists and is invoked to explain observations of coherent wave emissions (e.g. ‘lion roars’) in space plasmas. Oscillitons and periodic wave packets propagating obliquely to the magnetic field also exist although in this case the system becomes much more complicated, being described by four coupled differential equations for the amplitudes and phases of the transverse motion of the electrons and protons.