<p>Extreme situations, such as high density, low temperature, and strong magnetic-field dominance, may be pivotal in the nonlinear wave dynamics of the pulsar magnetosphere and are essential for interpreting various high-energy astrophysical phenomena. This theoretical study investigates weakly nonlinear fast magnetosonic wave structures in an electron-ion Fermi plasma relevant to the pulsar magnetosphere environment. A quantum magnetohydrodynamic model comprising inertialess degenerate electrons and nondegenerate ions, incorporating the effects of Fermi pressure and the quantum Bohm potential, is adopted for this purpose. Employing the reductive perturbation technique, a standard (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation is derived, and through the Hirota bilinear method, lump and multisoliton solutions are obtained. The findings reveal that the magnetic field strength, density and temperature significantly affect the fast magnetosonic lump and multisoliton structures in the pulsar magnetosphere. The investigation is further extended to study the modulational instability and rogue wave pulses in the low carrier wave frequency regime by obtaining a nonlinear Schrödinger equation from the KP equation. The magnetic field strength is seen to enhance the growth rate and the domain of modulational instability. However, it reduces the amplitude of the rogue wave pulses. Our investigation could provide momentum for space explorations aimed at studying the pulsar magnetosphere.</p>