  We study countably infinite Markov decision processes with B\"uchi
objectives, which ask to visit a given subset of states infinitely often. A
question left open by T.P. Hill in 1979 is whether there always exist
$\varepsilon$-optimal Markov strategies, i.e., strategies that base decisions
only on the current state and the number of steps taken so far. We provide a
negative answer to this question by constructing a non-trivial counterexample.
On the other hand, we show that Markov strategies with only 1 bit of extra
memory are sufficient.
