According to Eagle-Gazette-

Not every Ohio employer is mandated to abide by the state’s new smoking ban - which went into effect Dec. 7 - and the United States Post Office is one of them. The post office is a federal employer, so some state rules that apply to other businesses, such as the smoking ban, don’t apply to the employer, according Steve Rennick, lead safety specialist at the USPS Columbus district office.”The state can’t come in and enforce their rules in a federal facility,” Rennick said. “We have our own no smoking policy for customers and employees and will continue to enforce that.” Moore, a former smoker and 23-year Postal Service employee, hopes Lancaster Post Master Dan Miller takes initiative to do away with the smoking area, especially since many nonsmokers walk through the designated area.”The majority of (employees) don’t smoke and don’t want smoking there,” said Moore, who quit smoking one year ago.

From the ELM:

882 Policy
882.1 Buildings
Smoking is strictly prohibited in all buildings or office space (including service lobbies) owned or leased by the U.S. Postal Service. There will be no indoor smoking permitted by any occupant of such space. Local managers, with input from employee representatives, may decide whether or not to permit smoking in designated outdoor locations on Postal Service property.

882.2 Vehicles
Smoking is prohibited in any General Services Administration interagency fleet management system vehicles.