The Vatican City State did this between 1929 and 1984, indeed for all aircraft. The Lateran Treaty of 1929, between the Holy See and the Kingdom of Italy, said (Article 7, paragraph 3):
In conformità alle norme del diritto internazionale, è vietato agli aeromobili di qualsiasi specie di trasvolare sul territorio del Vaticano.
In accordance with the provisions of International Law, it is forbidden for aircraft of any kind whatsoever to fly over the territory of the Vatican.
The revision of 1984 did not preserve this provision (see its Article 13(1)). The present situation is that Vatican airspace, managed by the Italian national authorities, is subject to flight restrictions preventing low-flying aircraft and drones. So you cannot legally buzz St Peter's Basilica in your helicopter, but you could fly at a higher altitude in the normal way. (The restrictions are for anything below 3500 feet, and for comparison St Peter's is about 450 feet tall.)
There are obvious exceptions for papal or other official travel, and Popes have used their own helicopters to travel in and out of the Vatican since 1976. Since 2015, the helipad has also been made available for emergency use by medical helicopters from a nearby children's hospital. That demonstrates the extent to which it is not available for routine travel. Therefore, I think that "flying there for business purposes, flying domestically, or leaving the country by helicopter" are all forbidden, unless you are the Pope or have his permission.
You also cannot land or take off in a fixed-wing aircraft, but that's largely a matter of geography, not law: there is no runway.