Mary Elizabeth Frazier, 35, was being held in the Chatham County jail on an aggravated assault charge. Police said she also underwent a mental evaluation because she has a history of mental illness.
Her daughter, Diamond Frazier, was being treated at a local hospital. Police said the girl was in critical but stable condition Monday afternoon.
Savannah-Chatham County Police spokeswoman Judy Pal said investigators do not believe the shooting early Monday was an accident, though she declined to give a suspected motive.
"We believe that she purposely shot her daughter," Pal said. "We have information from family members and from doctors that she has a a history of mental illness."
The mother and daughter were the only people in their home, a duplex in a public housing complex near downtown Savannah. Police recovered a handgun inside the home, Pal said.
A neighbor called police early Monday saying she thought a burglar had been shot inside Frazier's home, she said.
"I don't know if she called because she heard the gunshot or because the mother was screaming," Pal said. "But it turns out it was actually her daughter that was shot while the girl was sleeping."
It was the second time in recent weeks police were called to Frazier's home. According to a police report, Frazier told an officer Feb. 28 that she punched her boyfriend as she was holding a knife.
The boyfriend, Marlon Angelo Houston, told police he woke up early that morning and saw Frazier standing over him with a knife, the report said. He suffered an injured lip, but no stab wounds, and did not press charges.
The report says Frazier thought Houston and her daughter were letting people into her home that she didn't want there. Houston told police Frazier had been "having dreams about this stuff lately, then waking up acting crazy saying I've been in her daughter's room but I've been in bed with her (Frazier) the whole time."
Pal said police believe Houston was no longer living with Frazier when her daughter was shot Monday. It was not immediately known if Frazier had hired an attorney.
The shooting stunned Frazier's neighbors in the Edgar C. Blackshear Homes complex, who said Frazier mostly stayed inside when she was home but didn't appear to be a threat to her child.
Octavia Brown, who lives in a duplex directly behind Frazier's, said she a "pop" early Monday but didn't realize it was a gunshot.
"I don't understand it, I really don't," Brown said. "My kids could never do anything to me that would make me want to shoot them."
Nobody answered the door Monday at the home of the neighbor sharing a duplex with Frazier, who called 911 after the shooting.
Police confirmed that Frazier's 16-year-old son, Devon Frazier, died in 2007 when he drowned while swimming in the Ogeechee River.