Protests have continued in Turkey as demonstrators defied a government appeal to end deadly unrest in which two people have died.

On Tuesday, Deputy Prime Minister Bulent Arnic apologised for "excessive violence" against protesters trying to save a park in Istanbul, a campaign which acted as catatlyst for deeper tensions within the country.

A meeting between the deputy prime minister and some of the protesters took place on Wednesday in Ankara.

Police used tear gas and water cannon overnight and into Wednesday morning on hundreds of protesters, who ignored warnings to disperse in Istanbul, Ankara and the southeastern city of Hatay.

In Istanbul, thousands gathered peacefully at Taksim Square on Wednesday yelling defiance at Prime Minister Recep Tayyip Erdogan, who had earlier dismissed the protesters as "extremists" and "vandals".

At least 29 people were arrested in the coastal city of Izmir for encouraging rebellion over social media and tweeting "misleading and libellous information", according to the state-run Anatolia news agency.

Police are still searching for 9 others who will be arrested on the same charges.

Al Jazeera's Rawya Rageh, reporting from Ankara, said that protests on Wednesday seem to be relatively calmer and than the previous day and the arrests are reflective of government accusations that Twitter is being used for spreading lies.

The Taksim Solidarity group held a press conference after their meeting with Arinc where they requested allowing for freedom of expression and the banning of tear gas.

The group said their findings indicated that during the protests four people have lost their eyes due to gas canisters and 2,319 have been injured.

Al Jazeera's Andrew Simmons, reporting from Taksim Square in Istanbul, said some of the other demands being put include cancelling the demolition of Gezi park, the release of those arrested, the resignation of governors and police comissioners of Istanbul, Ankara and Hatay, and the opening of public spaces for protesting.

He said the protests have been mostly peaceful and violence is only sporadic, and described the mood at the square like that of a "picnic".

Warning strikes

On Tuesday, thousands of public-sector labour union workers called for a two-day warning strike, which has not yet begun, in solidarity with the anti-government protesters.

Members of the workers union said they will join the ongoing rallies on Wednesday.

Two of the country's deputy prime ministers have commented on the situation and have called for calm.

Speaking at the American-Turkish Council's annual conference in Washington, also attended by US Vice President Joe Biden, Deputy Prime Minister Ali Babacan said the government respects the right to non-violent protest and free speech, but that it must also protect its citizens against violence.

"There is a need for a strict distinction between the terrorist groups or illegal organisations versus citizens who are purely protesting on a non-violent basis," he said on Wednesday.

Biden said that only the Turkish people can solve the problems behind the protests.

"Turkey's future belongs to the people of Turkey and no one else. But the United States does not pretend to be indifferent to the outcome," he said.

'Excessive violence'

Earlier, Arnic apologised for "excessive violence" against protesters trying to save a park in Istanbul.

He said the state had acted harshly when it sent in police to clear environmentalists participating in a sit-in protest to save Istanbul's Gezi Park last week.

"At the beginning of the protests, the excessive violence used against people concerned about the environment was wrong. It was unfair and I apologise to those citizens," he said.

"The government has learnt its lesson from what happened. We do not have the right and cannot afford to ignore people. Democracies cannot exist without opposition."

However, Arinc refused to reconcile with those who joined the later anti-government demonstrations.

"The ones who caused the destruction to the public property and the ones who are trying to restrict people's freedoms, we do not need to apologise," he said.

Arinc was speaking after a meeting with President Abdullah Gul who, contrary to Erdogan, has praised the mostly peaceful protesters as expressing their democratic rights.

Yavuz Baydar, a political analyst for the Today Zaman newspaper, said the comments were significant and amounted to a rebuke for Erdogan from within his AK Party.

"Most probably [Gul and Arinc's comments] did not have the sanction of the prime minister," Baydar told Al Jazeera, noting that both were among the founding fathers of the AK Party."

"[Arinc] apologised clearly for the police brutality and the excessive use of tear gas, and he sent some ambiguous but clearly understood messages to Erdogan for his perceived arrogant speech."

Police crackdown

Police arresting at least 20 people in the Istanbul district of Dolmabahce on Tuesday evening, Al Jazeera's Gokhan Yivciger said.

A 22-year-old man died during an anti-government protest in a city near the border with Syria, with officials giving conflicting reports on what caused his death.

The Hatay province governor's office initially said Abdullah Comert was shot on Monday during a demonstration in the city of Antakya.

It backtracked after the province's chief prosecutor's office said an autopsy showed Comert had received a blow to the head and that there was no trace of a bullet wound.

Another man was killed in an accident with a taxi in Istanbul.

Despite facing the biggest challenge to his rule since he came to office in 2002, Erdogan left Turkey earlier on Monday on an official visit to Morocco, where he insisted the situation in his country was "calming down".

He earlier rejected talk of a "Turkish Spring" uprising by Turks who accuse him of trying to impose religious reforms on the secular state, and dismissed the protesters as "vandals", stressing that he was democratically elected.

Erdogan is in Algeria on the second day of a four-day official visit to north Africa and is expected to return to Turkey on Thursday.