A sniper fired repeatedly into a crowded schoolyard in south-central Los Angeles today, killing a 10-year-old girl and injuring at least 11 other people, three critically.

Almost four hours after the shootings began at 2:23 P.M., members of a Los Angeles Police Department paramilitary unit entered a home across the street from the schoolyard and found the body of a man they said was the sniper. They identified him as Tyrone Mitchell, 28 years old.

The police said he appeared to have killed himself. Before entering, the police fired at least 16 canisters of tear gas into the house.

Police officials said friends of Mr. Mitchell had told them he had been a member of the People's Temple and had been in Guyana the day that hundreds of followers of the Rev. Jim Jones died in mass murders and suicides. He reportedly escaped death because he was away from Jonestown on a trip to Georgetown, the capital of Guyana, when the deaths occurred in November 1978.

Police officials said his parents and at least two other close relatives were believed to have died at Jonestown.

The girl killed by the sniper today was identified as Shala Eubanks. The injured included nine children and two adults, according to Lew Ritter, a deputy Los Angeles police chief.

The shooting occurred as pupils were leaving for the day from the 49th Street School in the heart of a sprawling black and Hispanic community not far from the site of the 1965 Watts riots. Nearby Houses Evacuated

The police special weapons and tactics team evacuated homes around the house they surrounded. Police officials said no one had been inside the house except Mr. Mitchell. The police dispatched a team of officers specially trained to deal with barricaded suspects, but they failed to talk him into surrendering.

The police officials said that Mr. Mitchell had a reputation in his neighborhood as a habitual user of the drug PCP and that relatives had told them he was under the influence of drugs today. They said they had been unable so far to confirm the reports of Mr. Mitchell's connections with Jonestown.

Witnesses said there had been about 100 children in the schoolyard when the shooting began.

They said the shots rang out without any warning and children began to fall. Those who were not hit ran screaming in search of shelter. The injured were taken to hospitals nearby. Some Injured Are Listed

LOS ANGELES, Feb. 24 (UPI) - The critically wounded victims at Martin Luther King Jr. Hospital were listed as Anna Gonzalez, 8, Steve Gomez, 9, and Carlos Lopez Bernal, 24, who was walking by the school when the sniper started shooting.

The wounded children at Orthopaedic Hospital were identified as Myra Cruz, Iran Macias and Alicia Pena, all 10, and Latreece Williams, 11. A playground supervisor, Albert Jones, 50, was also listed there as wounded. Others injured were believed to have been treated and released.

Dessi McClain, who lives across the street from the school, said she first thought the shots were a car backfiring. ''But then it was so many shots,'' she said. ''The kids were running out of the school screaming and hollering. They were running across the street.

''The teachers were trying to get the kids out of the yard, to get them away from the school. The shooting was going on all the time. It just kept coming: Boom! Boom! Boom! Boom! Boom!''

Bea Ransome, the school's office manager, said: ''The kids were screaming, but they were quickly brought into the school building by adults in the yard. When I first heard shooting I didn't know what was going on. Then I saw dozens of kids down on the ground, but most of them weren't hurt, they were just trying to hide.''

Seven ambulances were sent to the school. Some of the injured students were flown out by police helicopter. Other students were evacuated from the premises safely.

A Fire Department paramedic, Jack Fry, said he and his partner drove their ambulance onto the schoolyard and started pulling children, some hiding behind trees and trash cans, many of them crying, into the vehicle to take them out of the line of fire.

''We pulled the kids, injured or not, into the ambulance,'' he said. ''We just wanted to get them safe. We didn't know what could happen.''

A neighbor of Mr. Mitchell's, Ronnie Love, who was on a high school track team with him, said the dead man had been ''wild all his life.''

''He had a short fuse,'' he added. ''He could go off at any time.''

Mr. Mitchell's uncle, Willie Mitchell, said his nephew had a ''history of problems.'' Deputy Chief Ritter described Mr. Mitchell as ''a well-known suspect in the Newton Street area.''

''Many of the officers have dealt with him before,'' Mr. Ritter said. ''He has a history of being involved with drugs, particularly PCP.''