The exercise asks to find a infinite shift space of finite type with entropy $\mathbf{0}$.
(If our alphabet is a finite set $\mathcal{A}$, a shift space of finite type is a closed subset of the full shift space of sequences of $\mathcal{A}$, that can be described as a subset of sequences which doesn't contain strings from some finite set of forbidden strings.)
I can think of a infinite shift space of finite type, with entropy $0$, which is not of finite type (which is also found here): take the alphabet to be $\{a,b\}$, and the shift space to be all bi-infinite sequences with one single $b$ surrounded by all $a$'s - it is a closed set, satisfying that the number of possible strings of length $n$ is $n+1$, and therefore the entropy is $$\lim_{n \to \infty}\frac{\log(n+1)}{n} = 0$$ But I couldn't think of an example of a finite type, please help.
Consider the $2\times2$ transition matrix $$ A=\begin{pmatrix} 1 & 1\\ 0&1\end{pmatrix} $$ (or its transpose). The spectral radius is $1$ but the corresponding topological Markov chain has infinitely many sequences.
You can obtain many other examples of a similar type: think of a graph that after a while gets pushed into a fixed symbol.