The shooter himself has been killed by police and several victims are now being treated at a local hospital.
Quist said about 10:30 a.m. local time that parents were able to get to the school to see their children.
He said the suspect has been identified, but his name has not been released pending notification of next-of-kin.
Students also were shot and wounded at a regional primary school, although police say no children are among the dead from what appeared to be "random", drive-by attacks.
Flint says he had been shooting "hundreds of rounds" from large magazines.
"That evolved to multiple victims and multiple shots at the school, the elementary school", he told Action News Now, a TV news channel in Northern California. Police recovered a semi-automatic rifle and two handguns from the scene.
"He said he saw bullets hit the truck in front of him, so he made his children get down and put his truck in reverse".
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"He's dead. He didn't make it", the witness said.
Law enforcement officers from a range of agencies have responded to the incident, including the Highway Patrol and the Federal Bureau of Investigation.
Karen Matray with the Tehama Department of Education tells KTVU the school is in a very rural area.
The small district school is staffed by just four teachers.
The nearby Rancho Tehama Airport was placed on lockdown.
Rancho Tehama is a community of about 3,500 northwest of Corning. California Gov. Jerry Brown issued a statement in the wake of the shooting: "Anne and I are saddened to hear about today's violence in Tehama County, which shockingly involved schoolchildren".