  By solving the Bogoliubov--De Gennes equations analytically, we derive the
fermionic zero-modes satisfying the Majorana property that exist in vortices of
a two-dimensional $s$-wave Fermi superfluid with spin-orbit coupling and Zeeman
spin-splitting. The Majorana zero-mode becomes normalisable and exponentially
localised to the vicinity of the vortex core when the superfluid is
topologically non-trivial. We calculate the energy splitting due to Majorana
hybridisation and identify that the $s$-wave Majorana vortices obey non-Abelian
statistics.
