In this paper we give an example of a finite positional game with perfect information and without moves of chance (a chess-like game) that has no Nash equilibria in pure stationary strategies. In this example the number n of players is 4, the number p of terminals is 5; furthermore, there is only one directed cycle. On the other hand, it is known that a chess-like game has a Nash equilibrium (NE) in pure stationary strategies if: (A) n 2; or (B) p 3 and (C) any infinite play is worse than each terminal for every player; or (D) each of n players controls a unique position; or (E) there exist no directed cycles. It remains open whether a NE-free chess-like game (with at least one directed cycle) may exist in each of the following four cases: (A’) n = 3; (B’) 2 p 4; (C’) n > 2, p > 3, and condition (C) holds; (D’) each of n players controls at most 2 positions. In our example n = 4, p = 5, condition (C) does not hold, and there is a player controlling 3 positions.