Not since the days of Zapatistas' Subcomandante Marcos has Latin America been so charmed by a rebel leader. This time, there is no ski mask, no pipe and no gun, just a silver nose ring.

Meet Commander Camila, a student leader in Chile who has become the face of a populist uprising that some analysts are calling the Chilean winter. Her press conferences can lead to the sacking of a minister. The street marches she leads shut down sections of the Chilean capital. She has the government on the run, and now even has police protection after receiving death threats.

Yet six months ago, no one had heard of Camila Vallejo, the 23-year-old spearheading an uprising that has shaken not only the presidency of the billionaire businessman Sebastián Piñera, but the entire Chilean political class. Opinion polls show that 26% of the public support Piñera and only 16% back his recently ousted Concertación coalition.

Wednesday saw the start of a two-day nationwide shutdown, as transport workers and other public-sector employees joined the burgeoning student movement in protest.

"There are huge levels of discontent," said Vallejo in a recent interview. "It is always the youth that make the first move … we don't have family commitments, this allows us to be freer. We took the first step, but we are no longer alone, the older generations are now joining this fight."

Elected as only the second female leader in the 105-year history of the University of Chile's student union, Vallejo, who is also a member of the Chilean Communist party, is the face of a movement the likes of which has not been seen since the last years of Augusto Pinochet in the 80s.

Hundreds of thousands of high-school and university students have refused to go to lessons since early June, calling for better and more affordable education and an end to a two-tier system that creates a few wealthy, elite colleges amid many underfunded public ones. Vallejo has organised several cacerolazos – protests in which participants bang pots and pans. Some demonstrations have turned violent.

"We don't want violence, our fight is not versus the police or to destroy commercial shops … our fight is to recover the right to education, on that we have been emphatic and clear," said Vallejo as she stood outside the presidential palace.

The government has rushed out a number of initiatives to try to head off protests, promising to amend Chile's constitution to include a guarantee of quality education and cutting interest rates on student loans from 6.4% to 2%. But the promise of an extra 1.9 trillion Chilean pesos (£2.5bn) in education spending has done little to quell the uprising. Few analysts believe the students will back down despite a heavy police presence at recent demonstrations.

As she spoke, Vallejo was surrounded by students laying out a huge peace sign made up of hundreds of empty teargas canisters that had been used against students.

"Here we have more than 50m pesos' worth of teargas bombs," said Vallejo. "Imagine how much was used on the regional or the national level? This is unacceptable, we want to reiterate our demand that we made to the minister of the interior that he step aside."

Tatiana Acuña, a government official in the ministry of culture, was recently fired for suggesting that the assassination of Vallejo would end the protests. On Tuesday, Chile's supreme court ordered police protection for the student leader.

Vallejo has become a cult figure – with odes on YouTube and predictions that her charisma may well catapult her into Chilean politics. "We are all in love with her," said the Bolivian vice-president, Álvaro García Linera.

At a recent gathering of Bolivian youth leaders he urged students to follow the example of the youth movements in the rest of South America. "You need to talk about what is happening in Argentina, Brazil or Chile, where there is a young and beautiful leader, who is leading the youth in a grand uprising," said García Linera.

Vallejo said on the subject of her looks: "You have to recognise that beauty can be a hook. It can be a compliment, they come to listen to me because of my appearance, but then I explain the ideas. A movement as historical as this cannot be summarised in such superficial terms.

"We do not want to improve the actual system; we want a profound change – to stop seeing education as a consumer good, to see education as a right where the state provides a guarantee.

"Why do we need education? To make profits. To make a business? Or to develop the country and have social integration and development? Those are the issues in dispute."