France stepped up security across the country on Tuesday, deploying extra troops to towns and cities after a series of bloody attacks on the public.

The additional measures in a country already on high alert against a possible terrorist threat followed three days in which three separate incidents left one person dead and more than 20 injured, including three police officers.

On Tuesday morning, French prime minister Manuel Valls announced the deployment of hundreds of extra troops and increased patrols over the holiday period by armed police and gendarmes.

Within hours, the increased security was in evidence across the French capital, with soldiers at the city’s most famous landmarks, transport hubs and popular shopping areas, including the Champs Elysées, where the public were deemed to be at risk from random attacks

The sense of threat was not helped by mixed messages from France’s politicians.

On Monday, Valls warned of the menace from Islamist radicals and “lone wolf” extremists returning from waging jihad abroad. “We have never experienced such a great danger of terrorism,” Valls said.

A day later, after holding a crisis meeting at Matignon, the prime minister’s official residence, and announcing that between 200 and 300 extra soldiers would be deployed, Valls sought to calm public fears, calling for “sangfroid and vigilance”.

He confirmed his belief that the three incidents were not linked. However, he said he understood the “very real and legitimate concerns” caused by the succession of events.

“In a great democracy like ours, the best response is to continue to get on with our lives calmly. The words vigilance and unity must be the order of the day,” Valls told Europe 1 radio.

“We should not underestimate these acts. In moments of crisis, there shouldn’t be arguments about what is the most appropriate response.”

The additional troops bring the number patrolling France to 780, said Valls.

The first of the attacks came on Saturday, when Bertrand Nzohabonayo, a former rapper from Burundi who had converted to Islam, who reportedly cried “Allahu Akbar”, attacked police officers with a knife before he was shot dead in Joué-lès-Tours near the city of Tours.

A day later, a 40-year-old man ploughed into pedestrians in Dijon, injuring 13 people. The man, who had been admitted to a psychiatric hospital on more than 150 occasions, targeted groups of people at five locations in the city during a rampage that lasted at least 30 minutes, in what investigators said appeared to be a premeditated attack. He is now in police custody and was “put under official investigation” for attempted murder, the equivalent of being charged.

On Monday evening, a van driver deliberately drove into a Christmas market in Nantes in western France, injuring 10 shoppers, one of whom – a 25-year-old man – died in hospital.

The driver, aged 37, who police said was suffering from “psychological and family difficulties” and “alcohol-related problems”, ploughed a white Peugeot van into a group of shoppers enjoying mulled wine at the market in a pedestrian area of the city. He then pulled out a knife and stabbed himself several times in the chest. He was taken to hospital, where his condition was described on Tuesday as serious but not life-threatening. He was also charged with murder and attempted murder.

Investigators said they found a notebook containing “confused ramblings” suggesting he had psychological difficulties. The assailant had a criminal record for theft in 2006 and vandalising a vehicle in 2008, according to police sources.

France’s authorities say there appears to be no connection between the three attacks, and only one has been linked to a suspected Islamist fundamentalist.

“We cannot speak of an act of terrorism,” Brigitte Lamy, the Nantes public prosecutor, said of the attack on the Christmas market. “It appears to be an isolated case.” She added that there had been no religious or other claims made. “We need to verify this, but it appears to be the same kind of attack as that which took place in Dijon.”

Valls and interior minister Bernard Cazeneuve went to Nantes on Tuesday afternoon to “support the victims”.

Hollande, who is on a foreign visit to the French overseas territory of Saint-Pierre-et Miquelon, an Atlantic island off the coast of Canada, had ordered Valls to organise the emergency meeting of the interior, justice, defence and social affairs ministers. They were joined by the chiefs of the gendarmerie and national police service.

Speaking from Saint-Pierre-et Miquelon, Hollande told reporters: “We cannot give in to panic, to confusion to fear.”