Unknown attackers threw explosives and shot demonstrators Saturday during a rally in Trat province's Khao Saming district organized by the People's Democratic Reform Committee, the BBC reported.
The protesters were calling on Yingluck to resign.
Police said a 5-year-old girl standing at a noodle stall when the attackers opened fire was shot in the head and died at the scene.
At least 30 other people were injured in the attack, including a child left in critical condition, the BBC said.
No group had claimed responsibility for the attack.
Meanwhile, two people -- a 12-year-old boy and a woman in her 40s -- died Saturday when a bomb exploded at an anti-government rally near a shopping mall in the Ratchaprasong area of Bangkok, the Erawan Emergency Center reported. The explosion injured 22 people, including two who were in critical condition, CNN reported.
Yingluck issued a statement Sunday night saying the government "will not tolerate terrorism."
"I would like to ask all sides of the political divide that we may see things differently and there are many ways to express those differences," she said. "But the use of violence that lead to deaths are not the civilized way of the living."