  In a $p$-wave Kitaev model, the nearest neighbor pairing term results in the
formation of the Bardeen-Cooper-Schrieffer (BCS) pair in the ground state. In
this work, we study the fermionic condensation of real-space pairs in a
$p$-wave Kitaev model on a square lattice with a uniform phase gradient pairing
term along both directions. The exact solution shows that the ground state can
be expressed in a coherent-state-like form, indicating the condensation of a
collective pairing mode, which is the superposition of different configurations
of pairs in real space. The amplitudes of each configuration depend not only on
the size but also on the orientation of the pair. We employ three quantities to
characterize the ground state in the thermodynamic limit. (i) A BCS-pair order
parameter is introduced to characterize the phase diagram, consisting of gapful
and topological gapless phases. (ii) The particle-particle correlation length
is obtained to reveal the polarity of the pair condensation. In addition, (iii)
a pair-pair correlator is analytically derived to indicate the possessing of
off-diagonal long-range order. Our work proposes an alternative method for
understanding fermionic condensation.
