* N. Caucasus leaders say region plunges into extremism
* Medvedev assures full support to combat insurgency
(Adds clashes with armed militants, paras 7-8)
SOCHI, Russia, Aug 28 (Reuters) - Leaders from Russia's turbulent North Caucasus told President Dmitry Medvedev on Friday that an Islamist insurgency had permeated all spheres of society and they were struggling to combat it.
Ingushetian regional head Yunus-Bek Yevkurov, still limping from a suicide bomb attack in June, said Islamic militancy would be impossible to fight without greater support from the Kremlin.
"(It) has permeated all facets of life in society. Today...(it) presents a serious threat to peace and order in the republic and to the region as a whole, it's impossible not to feel it," said Yevkurov.
As well as the immediate threat of destabilisation in the entire North Caucasus, the Kremlin is worried radical Islamism may spill over into other regions in Russia, which is home to some 20 million Muslims.
Medvedev met over two dozen leaders, Muslim clerics and North Caucasus experts in the Black Sea resort of Sochi, where he reassured them of Moscow's support for the region.
Islam's holiest month of Ramadan has been marked in Russia's south by a wave of suicide bomb attacks and armed assaults on police and security forces in Chechnya, where Russia has fought two separatist wars, and neighbouring Ingushetia and Dagestan.
As national television channels showed Medvedev meeting with North Caucasus leaders in prime time, news agencies carried reports of intensive shootouts between security forces and gunmen in the regional republic of Kabardino-Balkaria.
"According to preliminary data, three members of illegal armed formations were killed," RIA bew agency quoted police as saying. It said there were no losses among security forces.
Chechen leader Ramzan Kadyrov, an ex-rebel turned Kremlin loyalist, told Medvedev a new strategy must be found.
"They kill, they blow things up, they are devils... We know we are making mistakes."
"We must do all we can to achieve a spiritual victory".
Agreeing, Medvedev told them: "It is absolutely essential to ensure full support for the Islamic leaders, the muftiat, those who serve in the Caucasus."
"Without consolidating the authority of the Islamic leaders we will be unable to deal with the problems that exist."
Yevkurov said religious organisations were themselves divided and "are too far away from the authorities", alluding to previous comments that local law enforcement agencies must be changed.
"Any of our attempts to resist this criminal underground are doomed to failure if we do not have our own ideology. But how will we deliver it?" he said.
Medvedev also proposed a Muslim television channel and controls on access to Islamic education abroad as ways of tackling Islamist insurgency in the region.
"There indeed must be control... Unfortunately these people are returning ... (and) bring back unorthodox views on Islam," he said, echoing complaints from Yevkurov and Kadyrov who say many Islamic schools in Arab countries spread radical teachings.
"We failed to take timely action to take the situation under our control, while those foreign emissaries grabbed the initiative," Yevkurov said. (Writing by Amie Ferris-Rotman; editing by Ralph Bouilton)
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