Markov property implies strong Markov property when state space is countable

Question

In general it cannot be said that a Markov process must satisfy the strong Markov property (see for example A Markov process which is not strong Markov process (follow up 2) ).

But what if the state space is countable/discrete? Is it true that a continuous-time Markov process taking values on $\mathbb{Z}$ must satisfy the strong Markov property?