  We consider the Fleming--Viot particle system associated with a
continuous-time Markov chain in a finite space. Assuming irreducibility, it is
known that the particle system possesses a unique stationary distribution,
under which its empirical measure converges to the quasistationary distribution
of the Markov chain. We complement this Law of Large Numbers with a Central
Limit Theorem. Our proof essentially relies on elementary computations on the
infinitesimal generator of the Fleming--Viot particle system, and involves the
so-called $\pi$-return process in the expression of the asymptotic variance.
Our work can be seen as an infinite-time version, in the setting of finite
space Markov chains, of results by Del Moral and Miclo [ESAIM: Probab.
Statist., 2003] and C{\'e}rou, Delyon, Guyader and Rousset [arXiv:1611.00515,
arXiv:1709.06771].
