Let G be a profinite group in which every centralizer
C
G
(
x
)
(
x
∈
G
)
{C_G}(x)\;(x \in G)
is either finite or of finite index. It is shown that G is finite-by-abelian-by-finite. Moreover, if, in addition, G is a just-infinite pro-p group, then it has the structure of a p-adic space group whose point group is cyclic or generalized quaternion.