  We show that free Burnside groups of sufficiently large odd exponent are
non--amenable in a certain strong sense, more precisely, their left regular
representations are isolated from the trivial representation uniformly on
finite generating sets. This result is applied to the solution of a strong
version of the von Neumann -- Day problem concerning amenability of groups
without non--abelian free subgroups. As another consequence, we obtain that the
above--mentioned groups are of uniform exponential growth.
