You can set import/export permissions at the Group level. So if you create a new group just for that one customer
Backend > Customers > Groups > Create New
and then edit the Group "Lists" Tab settings you can set import to no.
Then add that customer to that group. Keep in mind that anything you set at Group level overrides the Backend > Settings > Customer settings so make sure you edit the other settings for that Group to what the customer is allowed to do such as quotas etc.