By Thomas Grove and Alissa de Carbonnel

SLAVIANSK/LUHANSK Ukraine (Reuters) - Pro-Russian separatists poured scorn on peace overtures from Ukraine's new president Petro Poroshenko on Saturday as fighting rumbled on in the east of the country.

Taking the oath of office in Kiev, Poroshenko appealed to the rebels to lay down their arms, offering peaceful dialogue and immunity from prosecution to "those who don't have blood on their hands".

But rebel spokesmen in the self-styled Donetsk People's Republic, which has declared independence from Ukraine and wants to unite with neighboring Russia, told Reuters the fight would continue.

"What they really want is one-sided disarmament and for us to surrender. That will never happen," said a top separatist official, Fyodor Berezin. "As long as Ukrainian troops are on our soil, I can see that all Poroshenko wants is subjugation," he said by telephone from Donetsk.

Since Poroshenko's election on May 25, government forces have stepped up what they call an 'anti-terrorist' campaign against the Russian-speaking separatists in the east.

The rebels have fought back, turning parts of the east into a war zone. On Friday they shot down a Ukrainian army plane and killed a member of the interior ministry's special forces in the separatist stronghold of Slaviansk.

Fighting continued around Slaviansk on Saturday and smoke could be seen rising above the surrounding forests.

Ukrainian armored personnel carriers and military transport vehicles lined the road leading into the city, and soldiers behind concrete blocks and sand bags trained their machine guns on cars and buses driving out.

Inna, 38, was leaving by foot with her mother and grandmother, carrying bags with food, water and clothes.

"All you hear is shelling and bombing. Yesterday entire houses burnt down. We've been hiding in the cellar for three days and we finally decided to leave. There is no water or electricity," she said.

In his speech, Poroshenko said the government was prepared to talk to peaceful citizens - "clearly not with gunmen and other scoundrels" - and would offer a safe corridor for fighters who had crossed the border from Russia to go home.

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But his appeal appeared to fall on deaf ears.

"We have reached the point of no return," said Andrei Sukhanov, commander of the separatist Kaskad (Cascade) militia, manning a road block in Slaviansk.

PROTEST IN LUHANSK

While the government tightens its grip around Slaviansk, now encircled by the army, it appears to be losing ground in its easternmost region of Luhansk, where border guards have fled several bases after coming under attack.

Some 200 people protested against the presidential inauguration in the center of the city, some laying flowers on the sun-baked sidewalk in memory of eight people killed on Monday.

Residents say they died in an air strike from a Ukrainian plane which blew a hole in the regional administration building. The Ukrainian military denied this, blaming a misfire by separatists.

Broken glass and plaster crunched under the feet of the demonstrators.

"Our government is doing America's bidding. Poroshenko, we appeal to you to stop this - do you really want bloodshed?" shouted a red-haired woman to applause.

Frightened by the air-raid warning sirens that ring out over the city at night, Dmitry Grib, 20, said he was leaving for Moscow.

"I came to take a quick look before leaving," Grib said. "I don't trust him (Poroshenko). I didn't elect him."

But Olga Polovinka, who works as a medic in a charity for the homeless, said she was going nowhere. "This is our land. Why should we leave?" the 75-year-old demanded.

Separatist leader Valery Bolotov, governor of the self-proclaimed "Luhansk People's Republic", was emphatic in his rejection of Poroshenko and Ukrainian rule.

"The Ukrainians have made their choice and they must live with it. As for our republic, we have no diplomatic relations with Ukraine," he told journalists, wearing combat fatigues in a conference room hung with crystal chandeliers.

"Today Ukraine got a new president and now the blood of our people and of Ukrainians will lie on his conscience."

(Writing by Mark Trevelyan; Editing by Stephen Powell)