Hong Kong police have fought running battles with protesters trying to break a security cordon around a university in the city, firing teargas both at activists trying to escape the besieged campus and at crowds trying to reach it from outside.

Police have said the demonstrators inside Polytechnic University had no option but to come out and surrender.

The sprawling campus has been occupied by demonstrators since last week, and has become the focus of the most prolonged and tense confrontation between police and protesters in more than five months of unrest in the semi-autonomous city.

Hundreds of protesters, including secondary school students, have been trapped inside for more than 24 hours, after clashes on Sunday during which protesters launched petrol bombs and shot arrows at police, who threatened to use live rounds.

Police said they had allowed Red Cross volunteers into the university to ferry out injured protesters but said the rest had no option but to give themselves up. “Other than coming out to surrender, I don’t see, at the moment, there is a viable option for them,” Cheuk Hau-yip, regional commander of Kowloon West district, told a press conference, adding that police had the ability and resolve to end the standoff peacefully so protesters should not try their luck.

Parents of some of the activists trapped inside the university gathered in front of police cordons on Monday night with signs that read “Save Our Kids”, while hundreds of other supporters poured into the streets around the campus to try to break the police blockade. Police used teargas and water cannon to keep them at bay.

When a group of protesters tried to escape from the campus, police fired teargas and rubber bullets at exits, preventing them from leaving. When another group attempted to flee later on, hiding under umbrellas and shields made from scraps, officers fired further rounds of teargas and deployed a water cannon, engulfing the area in smoke. Several protesters were arrested.

Robin Brant 白洛宾 (@robindbrant) 1345 group of protestors make a run for it, escaping along highway by #hongkong Polytechnic. Around 100 run and climb over barriers. Police fire heavy tear gas. Several protestors detained. pic.twitter.com/T0Qiee4mwa

The game of cat and mouse followed a night of mayhem in the Chinese-ruled city in which roads were blocked, a bridge was set on fire and a police officer was shot by a bow and arrow.

Some protesters abseiled off a footbridge to a road below, where they were met by motorbike riders helping them flee. It was unclear whether they got away safely.

Michael Zhang 張雨軒 (@YuxuanMichael) This was probably the most surreal thing I have ever witnessed in the Hong Kong protests. Protesters just attempted a daring escape through a bridge at Polytechnic University. Volunteers on motorbikes came in drives to drive them out asap. Police fired teargas. #HongKongProtests pic.twitter.com/huhSo3Mxo9

People have been hiding in buildings throughout the campus, said Seze Li, a 26-year-old protester from inside the university. “It’s a disaster. Everyone is running around, looking for exits. We heard the protesters [trying to break the siege] are coming. We are just waiting for them,” she said. Some people have escaped by climbing out of the building, she said, “but not everyone can do that”.

Democratic lawmaker Hui Chi-fung told Reuters: “The police might not storm the campus but it seems like they are trying to catch people as they attempt to run. It’s not optimistic now. They might all be arrested on campus. Lawmakers and school management are trying to liaise with the police but failed.”

Hong Kong’s chief executive, Carrie Lam, posted a statement on her Facebook page about the crisis, her first remarks about it since it began almost 36 hours ago. Lam criticised the protesters for shooting arrows at police, as well as throwing bricks and petrol bombs. “Police have many times made appeals. Those inside the campus should listen to police without delay,” she said.

Earlier, the university’s president, Jin-Guang Teng, had urged protesters to leave, saying the police had agreed to a ceasefire on the condition that protesters stopped their attacks but police then fired on demonstrators who tried to leave.

Facebook Twitter Pinterest A screengrab from Google Maps showing road closures around Hong Kong’s Polytechnic University on Monday. Photograph: Google Maps

By mid-afternoon local time, about 300 to 400 people were left in the university, according to Tang Siu Wa, 41, a volunteer on the campus. Asked what they planned to do, she said: “They are 20-year-old kids. They don’t have plans. Everyone is nervous.”

Tang said the group was exhausted and faced dwindling supplies. Some peaceful protesters wanted to leave and others wanted to stay, she added. “People are getting tired but they don’t want to surrender.”

Journalists have not been allowed near the university.

The intensifying violence came as local media reported that district council elections may not be held this Sunday as scheduled because of the demonstrations.

Cancelling or postponing polls is likely to worsen tensions: some protesters have been demanding that the government promise to hold the elections, seen as one of the last formal venues people have for expressing their views.

In another development, Hong Kong’s high court ruled that a ban on face masks implemented by the government was unconstitutional. The ban made wearing any facial coverings during public assembly punishable by prison time and fines.

Jessie Pang (@JessiePang0125) Police fire rounds and rounds of tear gas again at every exit they try to leave. The protesters are forced to trap in PolyU. #antielab #hongkongprotests pic.twitter.com/3BJv1QoV0O

Police had previously issued a statement ordering everyone inside the university to drop their weapons, remove their gas masks and leave. “The rioters are hereby warned to stop their unlawful acts,” the police said.

Representatives of the university’s student union posted a statement on Facebook saying police had blocked all exits since Sunday night. The union said several protesters were in need of medical help, including three people with eye injuries and about 40 experiencing hypothermia after being hit by water cannon.

Q&A Why have universities become a flashpoint in the Hong Kong protests? Show Hide Hong Kong’s protest movement has evolved throughout the five months it has raged in the harbour city. Its latest phase is being played out on Hong Kong’s university campuses — traditionally sites of political activism — some of which have been transformed into makeshift fortresses by demonstrators in the past fortnight. The immediate trigger for the campus confrontations appears to have been the death on 8 November of a Hong Kong University of Science and Technology student. Since then, several campuses have been barricaded by students, some of who are using footbridges or or near the campuses to block roads. At least three campuses are blockaded, including the Hong Kong Polytechnic University, close to the cross-harbour tunnel — a key artery for traffic in the city, and one that authorities will be determined to keep open.

The campus protests have been desperate: activists are using petrol bombs, bamboo poles and other weapons including javelins and bows and arrows. Observers have told the Guardian the shift to campuses represents a major escalation. Many of those occupying the campuses are students or alumni, and until recently, riot police have refrained from entering universities. “The university is the home turf of the students,” Ho-Fung Hung, a professor in political economy at Johns Hopkins University, has told the Guardian. “There is this notion of academic freedom and the university as a bastion of free ideas, this notion of autonomy. To people, this should not be breached by authorities.”

“Because most of the emergency relief team and first-aiders have been arrested and taken away, there are insufficient resources and personnel within campus to treat the injured,” the statement said, calling the situation “a severe humanitarian crisis”.

In other neighbourhoods, police fired teargas and water cannon at protesters and other supporters who had occupied streets and built barricades in an attempt to divert police resources overnight on Sunday. Dozens were seen being arrested when they returned on Monday. Volunteers with cars blocked roads to slow the police.

Hong Kong is experiencing its most serious political crisis in decades after the government attempted to push through a controversial extradition bill that would have allowed suspects to be sent to mainland China, seen by many as another move to extend Beijing’s control over the city, which Britain returned to its control in 1997.

Protests over the now withdrawn bill pose a direct challenge to China, which governs Hong Kong under the “one country, two systems” framework.

The US secretary of state, Mike Pompeo, urged the city’s government to address the protestors’ concerns, calling on the Chinese Communist party to honour its commitment to uphold freedoms and liberties.

“The Hong Kong government bears primary responsibility for bringing calm to Hong Kong,” Pompeo said. “Unrest and violence cannot be resolved by law enforcement efforts alone. The government must take clear steps to address public concern.”

The European Union urged restraint on all sides, stressing that “any violence is unacceptable”.

“It is crucial that all sides exercise restraint and engage constructively in de-escalation efforts,” the EU’s foreign policy chief Federica Mogherini said in a statement.

Facebook Twitter Pinterest People are detained by police near the university in Hung Hom district on Monday. Photograph: Dale de la Rey/AFP via Getty Images

Earlier, the People’s Daily, the official mouthpiece of the Chinese Communist party, published an editorial on its front page saying there was no room for compromise: “What we are facing today is a struggle between safeguarding ‘one country, two systems’ and destroying it.

“On an issue involving national sovereignty and the future of Hong Kong, there is no middle ground and absolutely no room for compromise.”