“Today is a sad day not only for Madison, for our entire country,” Chief Barnes said. “We have to do a better job in our community.”

He added the police had not identified a motive in the shooting, and the suspect’s family was co-operating with the investigation. He said it is not yet clear how the attacker got hold of a firearm.

He named the alleged attacker as Natalie Rupnow, who also went by the name Samantha. She is believed to have died from a self-inflicted gunshot wound.

The official cause of death will be released by the Dane County Medical Examiner pending autopsy results.

Chief Barnes said that, to his current knowledge, police had not had any prior interactions with the alleged shooter.

Officers responded to a 911 call of a shooter at the Christian school around 11:00 local time (17:00GMT) on Monday. The attacker attended school before the shooting, Chief Barnes said.

The shooting was confined to a study hall with students in mixed grades.

Barbara Wiers, director of relations at the school, said the school had conducted active shooter training earlier this year and the information was “very fresh” for educators to put into practice on Monday.

She said while the school does not have a dedicated police officer, known as a school resource officer, the doors of all classrooms automatically lock and anyone wanting to gain entry to the campus must be buzzed in through the primary entrance.

Ms Wiers, who said she was teaching at the time of the attack, said students handled themselves “brilliantly”.

“They were clearly scared,” she said. “When they heard ‘lockdown, lockdown’ and nothing else, they knew it was real.”

Police say they found the shooter dead when they arrived at the school, along with a handgun. No officers fired weapons.

Police have not named any of the victims.

Chief Barnes said two students were in critical but stable condition in the hospital facing life-threatening injuries. Four others were taken to hospital with non life-threatening injuries, two of whom have since been released.

Authorities have appealed for witnesses who saw or heard the attack to come speak to police, and that they hope these accounts will shed light on the attacker’s motive.

“But that’s not something we want to rush. We’re not gonna interrogate students,” Chief Barnes said. “We’re gonna give them an opportunity to come in and tell us what they saw when they’re ready.”  He added that “eveyr child, every person in that building is a victim and will be a victim forever”.

The chief said officers responded to the school as they were undergoing training at a training centre for law enforcement located three miles away.

“What began as a training day became an actual day,” he said.