  In this paper, we discuss the stability of (restricted) Chebyshev centers in
few function spaces. For an extremally disconnected compact Hausdorff space $K$
and a finite dimensional Banach space $X$, we prove the existence of Chebyshev
centers of closed bounded subsets of the space of $X$-valued continuous
functions on $K$, $C(K,X)$ and that of compact subsets of $M$-ideals in
$C(K,X)$. It is also proved that the existence of restricted Chebyshev centers
is stable in the space of vector-valued bounded functions on an arbitrary set.
Furthermore, the stability of continuity properties of the Chebyshev-center map
of a Banach space $X$ in dependence on the continuity properties of the
Chebyshev-center map of $C(K,X)$ is studied.
