"Preliminary information is that the alleged shooter has taken his own life here at the scene," she said, adding that there was no longer any threat to the community.
Four people were transported to hospitals, one in critical condition with gunshot injuries and three others with various other injuries, she said.
Two more were treated at the facility itself by medical staff and released, while "multiple" others had walked in to hospitals nearby, she added.
"It is very heartbreaking and, you know, in the Indianapolis Metro Police Department, the officers responded, they came in, they went in and they did their job," Cook said.
"And a lot of them are trying to face this because this is a site that no one should ever have to see."
Television footage showed victims' families, friends and work mates, all with face masks on due to the pandemic and most holding their mobile phones, at a "family unification centre" set up by authorities in a nearby hotel.
At least 30 people have been killed in mass shooting incidents around the country in the last month.
A FedEx spokesman said the company was "deeply shocked and saddened" by the loss of its team members. He did not clarify if all those killed were FedEx employees.
"Our most heartfelt sympathies are with all those affected by this senseless act of violence. The safety of our team members is our top priority, and we are fully cooperating with investigating authorities," he added.
U.S. Congressman André Carson, a representative for Indiana, wrote on Twitter: "I am heartbroken by the mass shooting at the FedEx facility here in Indianapolis and praying for all affected by this tragedy.
"I am communicating with local authorities to get all details of the attack and my office stands ready to help everyone affected in any way we can."
(Reporting by Shubham Kalia, Nandakumar D and Akshay Lodaya in Bengaluru and Njuwa Maina; Editing by Raju Gopalakrishnan, Michael Perry, Muralikumar Anantharaman and Hugh Lawson) ((Shubham.Kalia@thomsonreuters.com;))