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


  [1]: https://www.jstor.org/stable/3532076#metadata_info_tab_contents