  The p-wave Cooper-pairing instability in superfluid $^{3}$He, characterized
by a parity-breaking excitation gap, is regarded as one of the most rich and
complex phenomena in physics. The possibility of a counterpart unconventional
p-wave ordering of interacting fermions, in which a Fermi surface spontaneously
breaks the parity symmetry, has been an open problem for many decades. Here we
identify the realization of the counterpart of p-wave superfluidity in
magnetism. We demonstrate a strong parity-breaking and anisotropic symmetry
lowering of spin-polarized and time-reversal symmetric Fermi surfaces in a
representative p-wave magnet CeNiAsO. As a direct experimental signature we
predict a large spontaneous anisotropy of the resistivity. Abundant and robust
realizations of the unconventional p-wave magnetism can be identified from
suitable non-relativistic crystal-lattice and spin symmetries, without
requiring strong correlations and extreme external conditions. This opens new
prospects in fields ranging from topological phenomena to spintronics.
