We know that if we have a Markov chain $(X_n)$ on an finite state space $S$ and with at least one recurrent state then the chain is irreducible and hence recurrent.
So my question is : If we have an irreducible Markov chain $(X_n)$ with one transient state then does that mean that the state space $S$ is infinite?
Yes. Recall that an irreducible chain is one where the entire state space $S$ is a single communicating class. In this case clearly $S$ is closed.
Further, recurrence/transience is a class property. So in an irreducible chain, either every state is recurrent or every state is transient.
It is a theorem that for a closed class $C$, then $C$ finite $\implies$ $C$ is recurrent (this is very closely related to the fact that you state).
So for an irreducible chain, applying the contrapositive of this theorem to the closed class $S$ gives: $S$ transient $\implies$ $S$ is infinite.