There is no such counterexample. Indeed, by [Theorem 1][1], any strong mixing sequence obeys the $0$-$1$ law. Also, any [$m$-dependent][2] sequence is obviously strong mixing, for any integer $m\ge0$, and hence must obey the $0$-$1$ law. 


  [1]: https://www.jstor.org/stable/3532076#metadata_info_tab_contents
  [2]: https://encyclopediaofmath.org/wiki/M-dependent-process