The set $\{x \in Q=[-1,1]^\mathbb{N}: \forall n: x_{2n} = 0\}$ is closed, has empty interior and is no $Z$-set in $Q$. I think also $\{0\} \times [-1,1]^{\Bbb N}$ works.

For more info on $Z$-sets in $Q$, see Infinite-dimensional Topology by Jan van Mill.

$A=\{0\} \times [-1,1]^{\Bbb N}$ works as an example, e.g.

For more info on $Z$-sets in $Q$, see Infinite-dimensional Topology by Jan van Mill.