Profinite groups with few conjugacy classes of p-elements

It is proved that a profinite group G has fewer than 2^{\aleph _0} conjugacy classes of p-elements for an odd prime p if and only if its p-Sylow p-subgroups are finite. (Here, by a p-element one understands an element that either has p-power order or topologically generates a group isomorphic to \mathbb {Z}_p.) A weaker result is proved for p=2.