ABUJA (Reuters) - Nigeria's military said on Monday it killed 35 members of the Islamist sect Boko Haram and arrested several during an overnight gunbattle in Damaturu, capital of northeastern Yobe state.

Boko Haram, which wants to carve out an Islamic state in northern Nigeria, has been blamed for more than 1,000 deaths since its insurgency intensified in 2010. The United States has designated three of Boko Haram's senior members as terrorists.

"The operation which lasted throughout Sunday night and continued into the early hours of Monday recorded fierce exchanges of fire between Joint Task Force (JTF) troops and the terrorists," said a statement from JTF spokesman Eli Lazarus.

"Thirty-five terrorists were killed in the fight while several others were arrested ... two JTF men were injured," said Lazarus.

Nigeria's military has been accused of using heavy-handed tactics in the past and previous operations targeting Boko Haram have resulted in civilian deaths.

Lazarus said the JTF had lifted 24-hour curfews imposed on Damaturu and the nearby town of Potiskum, both scenes of several deadly Boko Haram attacks this year.

A military crackdown appears to have weakened Boko Haram, whose militants have not reproduced the kind of large-scale, coordinated attacks they carried out earlier this year. At least 186 people were killed in the city of Kano during a wave of coordinated bomb and gun attacks over two days in January.

But almost daily shootings and bombings blamed on the Islamists have continued.

A suicide car bomber blew himself up outside a Catholic church in northern Nigeria on Sunday, killing at least two people and wounding 46, police said. Boko Haram has targeted churches in attacks.

The militants have made no public statement since government forces said they killed the sect's spokesman Abu Qaqa in a gunbattle in Kano on September 16.

Security analysts say Boko Haram has forged links with other jihadist movements expanding across West Africa like al-Qaeda in the Islamic Magreb, a group formed in Algeria and now based in northern Mali.

But apart from an attack on the U.N. headquarters in the Nigerian capital Abuja last August, Boko Haram's focus has been mainly on local targets. (Reporting by Isaac Abrak; Writing by Joe Brock; Editing by Ralph Gowling)