By Andrew Cawthorne and Diego Ore

CARACAS (Reuters) - Venezuela's jailed protest leader urged supporters on Friday to keep demonstrating peacefully against President Nicolas Maduro despite violence that has killed at least six people and rocked the OPEC member nation.

"I'm fine, I ask you not to give up, I won't," Leopoldo Lopez told his followers in a handwritten note passed to his wife at Caracas' Ramo Verde prison and then posted on the Internet.

The 42-year-old Lopez, a Harvard-educated economist and one of the few surviving relatives of Venezuelan independence hero Simon Bolivar, spearheaded protests against the socialist government that began at the start of February.

He surrendered to the military this week after an arrest warrant was issued, accusing him of instigating the violence.

At least six people have died, five from gunshots, and one run over by a vehicle, as the protests have degenerated into violence in Caracas and other cities around Venezuela, especially in the western Andean region.

The government puts the death toll at eight, including indirectly linked cases of a woman who had a heart attack and a judicial official who crashed his car swerving round a barricade.

Some 100 civilians have also been injured, and 37 members of the security forces, authorities say.

Both sides are blaming each other for murder and brutality.

The government says sharpshooters are appearing on the opposition side, and radicals are seeking to create chaos by smashing property, attacking police and blocking highways.

The protesters, mainly students, accuse Maduro of worsening repression. They say police are firing shots, allowing pro-government gangs to attack protesters and mistreating some detainees.

"To the police, soldiers, prosecutors and judges: do not obey unjust orders, do not become the face of repression," Lopez said in his note from prison.

"To the youth, to the protesters, I ask you to stay firm against violence, and to stay organized and disciplined. This is everyone's struggle."

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Having initially accused Lopez of crimes including murder and terrorism, authorities are now charging him on lesser counts of instigating arson, damage and criminal gatherings.

"WAR ZONE" IN WEST VENEZUELA

Once again, the worst trouble on Friday appeared to be in the western border town of San Cristobal, which residents are calling a "war zone," with running battles for days between students and security forces on barricaded streets.

Venezuela's military has moved into the town in force, residents say, with helicopters and planes flying overhead.

There was also trouble in Merida, another Andean town, and protesters blocked a few streets again in Caracas on Friday.

In the biggest challenge to Maduro's 10-month-old government, the protesters are demanding his resignation over Venezuela's rampant crime, inflation, shortages of basic products and alleged repression of opponents.

Demonstrators bang pots and pans from windows every night.

"I recommend they buy some stainless steel pots to last for a good 10, 20, 30 or 40 years," Maduro mocked them.

"Because the revolution is here for a long time!"

While the Caracas protests began in middle-class neighborhoods and are still strongest there, sporadic demonstrations have also spread to poorer areas.

Maduro says the protests are a pretext for a planned coup, similar to the short-lived ouster of his predecessor, Hugo Chavez, in 2002. There is no evidence the military, which was the decisive factor in 2002, may turn on Maduro now.

Oil minister Rafael Ramirez warned that fuel supplies to "zones under fascist attack" could be suspended in order to preserve security.

Maduro, a 51-year-old former bus driver and union activist, won the presidential election last April. About nine people, most of them government supporters, were killed then in post-election violence when the opposition disputed the result.

RAPPROCHEMENT WITH WASHINGTON

Maduro on Friday called on U.S. President Barack Obama to hold dialogue with his government and proposed the two countries restore ambassadors, a day after slamming U.S. President Barack Obama's comments on Venezuelan affairs.

"Obama says we should hold dialogue, then I call for a dialogue with you, President Obama, I call for a dialogue between revolutionary Venezuela and the United States and its government," Maduro said in an evening press conference.

The U.S. president on Thursday criticized Maduro's government for arresting protesters and urged it to focus on addressing the "legitimate grievances" of its people.

Venezuela on Monday expelled three more U.S. diplomats on charges of stirring up protests, an accusation the U.S. State Department called "baseless and false."

The two countries have not had full diplomatic relations since 2008. An attempt last year to restore ambassadors ended after Venezuela expelled three U.S. diplomats on charges of espionage and Washington responded in kind.

Bolivian President Evo Morales called on Obama and Colombia's conservative leader Juan Manuel Santos to stand up for Maduro after both called for dialogue in comments that Venezuela's government bristled at.

"If they are democratically elected presidents, they have to defend President Maduro because he's been democratically elected by his people," Morales said.

Local TV channels are providing almost no live coverage of the unrest, so Venezuelans are turning to social media to swap information and images. Falsified photos are also circulating.

Maduro has criticized CNN's coverage of the unrest, and the U.S.-based network said on Friday its reporters' credentials had been revoked. Information Ministry officials did not confirm that, saying CNN had been invited to a Maduro news conference later on Friday.

(Additional reporting by Javier Farias in San Cristobal, Daniel Wallis, Eyanir Chinea and Tomas Sarmiento in Caracas, and Daniel Ramos in La Paz; Editing by Kieran Murray, James Dalgleish and Ken Wills)