There are such materials! See chp. 11 in Girvin-Yang which gives a much better discussion than I am providing here. You could probably also look up 'Anderson insulator' online or in any other solid-state text book for a similar explanation.
The Einstein relation for conductivity is $ \sigma \sim \rho D $ where $\sigma$ is conductivity, $\rho$ is the density of states at the Fermi level, and $D$ is the diffusion constant. $\rho$ basically tells you how many electrons there are that can conduct and $D$ tells you how easily these electrons can move.
If $\rho=0$, there are no electrons that can move, so there is no conductivity. This is the case for gapped materials like band insulators.
On the other hand, there are materials where $D=0$ while $\rho \neq 0$. The only example I can think of is an Anderson insulator. My understanding is tenuous, but basically in systems with strong disorder, it is possible for electrons to get trapped in deep potential wells. There are electrons in the system, i.e. $\rho \neq 0$, but the potential makes it very difficult for them to move and $D\rightarrow0$, i.e. $\sigma\rightarrow 0$ even though $\rho \neq 0$. I think the fact that the potential is disordered is important since a very strong lattice potential would lead to a different type of metal-insulator transition, namely a Mott transition, where the limit of isolated atoms is approached. Like I said, my understanding of Anderson insulators is poor, but hopefully this answer helps! Feel free to read more about Anderson insulators and explain them to me.
Cheers,
Ty
