  The existence of a countably compact group without non-trivial convergent
sequences in ZFC alone is a major open problem in topological group theory. We
give a ZFC example of a Boolean topological group G without non-trivial
convergent sequences having the following "selective" compactness property: For
each free ultrafilter p on N and every sequence {U_n:n in N} of non-empty open
subsets of G one can choose a point x_n in U_n for all n in such a way that the
resulting sequence {x_n:n in N} has a p-limit in G, that is, {n in N: x_n in V}
belongs to p for every neighbourhood V of x in G. In particular, G is
selectively pseudocompact (strongly pseudocompact) but not selectively
sequentially pseudocompact. This answers a question of Dorantes-Aldama and the
first author. As a by-product, we show that the free precompact Boolean group
over any disjoint sum of maximal countable spaces contains no infinite compact
subsets.
