  We provide explicit examples of positive and power-bounded operators on $c_0$
and $\ell^\infty$ which are mean ergodic but not weakly almost periodic. As a
consequence we prove that a countably order complete Banach lattice on which
every positive and power-bounded mean ergodic operator is weakly almost
periodic is necessarily a KB-space. This answers several open questions from
the literature. Finally, we prove that if $T$ is a positive mean ergodic
operator with zero fixed space on an arbitrary Banach lattice, then so is every
power of $T$.
