The condition of the former Formula One world champion Michael Schumacher has improved slightly after doctors carried out a second operation to reduce the swelling on his brain following an off-piste skiing accident in the French Alps.

In a press conference at Grenoble hospital on Tuesday doctors said they had carried out a two-hour operation to reduce the haematoma on his brain. They said a scan after the procedure had showed a slight improvement, but they were unable to say the former racing driver was out of danger.

Gerard Saillant (L) and Professor Jean-Francois Payen leave a press conference at Grenoble University Hospital. Photograph: Alex Grimm/Bongarts/Getty Images

Schumacher spent his second night in hospital as doctors treating the former Formula One driver battle "hour by hour" to save his life after injuries sustained after falling while skiing in the resort of Méribel on Sunday.

Schumacher's wife, Corinna, and their two children are with him in hospital. So too is Gérard Saillant, a leading brain surgeon from Paris who had rushed to Grenoble on Sunday to be with his former patient and friend.

Doctors said the family had taken the "difficult decision" to go ahead with a second operation when a window of opportunity presented itself. The procedure took place at 10pm on Monday night and lasted two hours.

They said that following the operation there had been "some signs" that Schumacher's condition was "more under control now". While they could not say he was out of danger they had "gained a bit more time in the evolution" of his condition, doctors said.

Michael Schumacher is being treated for a head injury following a skiing accident. Photograph: Alex Grimm/Bongarts/Getty Images

Schumacher was airlifted to Grenoble on Sunday after falling heavily and hitting his head while skiing on an unmarked slope in the resort of Méribel.

It soon emerged that the accident, initially played down by a spokesman for the resort, had a devastating impact on Schumacher, who was in a coma by the time he arrived at Grenoble.

Doctors said the first operation, on Sunday, had been to eliminate extradural haematoma – outside the actual brain – to reduce pressure. A scan later showed that one of the haemotoma within the left of the brain was larger and accessible they said, and they could eliminate it without risk so they carried out the second operation.

"I want to assure everyone that all patients who aren't famous will be treated with the same amount of care," one of the doctors told the press conference.

Michael Schumacher on the slopes. Photograph: Olycom SPA/REX

The gravity of the retired champion's condition prompted figures from across the sporting world to express their shock at the accident and solidarity with a man regarded as one of the greatest racing drivers in Formula One's history.

The current world champion, Sebastian Vettel, a fellow German who has been dubbed Baby Schumi by his country's media, led a chorus of support and concern. "I am shocked and hope that he gets better as quickly as possible. I wish his family much strength," he said.

The British driver Jenson Button, another former world champion, said: "My thoughts are with Michael Schumacher at this tough time. Michael more than anyone has the strength to pull through this."

Angela Merkel, the German chancellor, said she was "extremely shocked along with millions of Germans".

According to one French media report, the blow to Schumacher's head was so fierce that the helmet cracked. That could not be independently confirmed, but prosecutor Patrice Quincy, based in Albertville, the home of the 1992 Winter Olympics, said: "The victim lost his balance and banged his head extremely violently on small rocks concealed by the snow."

Two ski patrollers arrived at the scene of the accident immediately, Quincy added, and Schumacher was taken by helicopter to the nearest hospital, at Moûtiers.

Schumacher is thought to have been skiing downhill between two marked pistes below the jagged peaks of the Dent de Burgin when he fell, hitting the right side of his head.

Investigators from the gendarmerie based in the resort of Bourg St Maurice were on the mountainside on Monday attempting to establish the exact circumstances of the accident, which occurred at an elevation of 2,100 metres.

After the impact Schumacher was conscious but "stunned", neurosurgeon Stephan Chabardes said. "He didn't respond to questions after the accident. He didn't have a normal neurological reaction," he told the news conference. By the time he was brought to the hospital at Grenoble, which has an internationally recognised trauma centre, Schumacher was in a coma.

Tony Belli, a consultant neurosurgeon and reader in neurotrauma at Birmingham University, said Schumacher's age, fitness and psychological strength would help his recovery. "He's got a lot of factors in his favour," he said. "He's more likely than not to make a good recovery, despite all the concerns that have been raised.

"From the description of his injuries, it will take weeks, even months to recover, but I do expect him to make a good recovery."

Outside the hospital, fans mingled with television crews as the wait for a further update went on into the night.

Dressed in a red Ferrari cap, jacket and T-shirt, Enzo Debar, 31, said he had come to the hospital "to be close to him, to do what is possible, as a fan. I hope he'll get better. I am sure he will get better, because he's a fighter."

Sharing his hope was Patrick Amatucci, from Grenoble, who had come down with his son. He described Schumacher, whom he had followed from his days with the Benetton team, as "a great, good man", with greatness in his blood.