* French, Chadians secure town of Tessalit in far northeast
* Malian soldiers open fire on paratroop base protesters
* Interim president condemns "fratricidal" army violence
* In north, first suicide bomber attacks army checkpoint
* France, allies want political deal, elections for stability (Recasts, adds presidential statement, EU trainers, Algeria arrests)
By Tiemoko Diallo and David Lewis
BAMAKO/GAO, Mali, Feb 8 (Reuters) - French forces hunting Islamist rebels in northern Mali parachuted into a strategic town and took control on Friday, but a suicide bombing further south and the killing of two civilians by soldiers in the capital Bamako raised fresh security fears.
In their hot pursuit of al Qaeda-allied militants in Mali's remote Saharan northeast, French special forces seized the town and airfield of Tessalit, about 50 km (30 miles) from the Algerian border.
It was the northernmost town secured so far by French and Chadian troops in their drive to flush retreating jihadist insurgents out of their hideouts in the Adrar des Ifoghas mountains, where they are believed to hold French hostages.
In a four-week lightning military intervention launched by Paris on Jan. 11, some 4,000 French troops backed by warplanes and helicopters have driven Islamist rebels out of the main urban centres of northern Mali into the mountains and desert.
But the speed of the offensive has opened up a potential security vaccum behind the fast-moving French forces, where liberated towns and areas are meant to be secured by the Malian army and a larger African force that is still deploying.
FIRST SUICIDE BOMBING
In an ominous sign of vulnerability, Mali's first reported suicide bombing since the arrival of French forces left a Malian soldier injured on Friday on the outskirts of Gao, recently retaken by French and Malian troops from the Islamists.
A soldier who witnessed the attack said a man on a motorbike approached a checkpoint from the town of Bourem. Ignoring orders to halt, he pulled up his shirt and there was a large blast.
The charred, twisted remains of the motorbike could be seen by the side of the road and shrapnel scarred a nearby wall.
The attack followed the expulsion by French-led forces of Islamist rebels from their desert strongholds of Gao, Timbuktu and Kidal.
Since the recapture of Gao, at least two landmine explosions on a main route to the town have killed several Malian soldiers.
In Bamako, government soldiers and police opened fire at a camp housing paratroopers loyal to deposed President Amadou Toumani Toure and their families, killing or wounding a number of civilians, witnesses said.
The government said two teenagers were killed and 13 other people wounded, but it offered no explantion for the incident.
The military coup last March that ousted Toure resulted in Tuareg rebels seizing the north of the country in a revolt later hijacked by Islamist radicals. Mali, a former French colony, is Africa's third-largest gold producer.
Malian government officers had initially reported a heavy gunbattle involving the pro-Toure paratroopers at the Djikoroni-Para base in western Bamako, saying they had defied orders reassigning them to other units at the frontline.
But witnesses at the base, including a Reuters photographer, said they saw the government forces open fire when the wives and children of the paratroopers protested, some throwing stones.
"The soldiers and gendarmes burst in and started shooting," Seydou Kone, a resident at the camp, told Reuters.
In an emotional appeal later on Friday to the country's army, Mali's interim President Dioncounda Traore condemned "this sad spectacle of fratricidal shooting between you".
"The Malian army has certainly got better things to do than what we've seen today," he said bitterly.
The Bamako shooting highlighted tensions and splits that have lingered in Mali's armed forces since the 2012 coup that plunged the previously stable West African state into chaos.
The incident pointed to weaknesses in the Malian state that could undermine the rapid gains made by the French military operation, which is backed by the United States and Europe as a defence against Islamist jihadists threatening wider attacks.
An advance party of European Union military trainers arrived in Mali on Friday with the challenging task of raising the combat capacity of the Malian army.
Residents of the capital expressed frustration.
"I don't understand how at a moment when French and African forces are here to fight our war in our place ... Malian soldiers, instead of going to fight at the front, are fighting over a stupid quarrel," said one west Bamako resident, Assa.
Malian and other troops from a U.N.-backed African force expected to number 8,000 are still being deployed in the desert towns formerly held by the Islamists and are due to provide security behind the French forward lines.
Neighbouring states are also cooperating. The Algerian army captured four heavily armed militants near the borders with Mali and Libya, a security source said on Friday.
NEED FOR POLITICAL REFORMS
To accompany the military offensive, France and its allies are urging Mali's authorities to open a national reconciliation dialogue that addresses the pro-autonomy grievances of northern communities like the Tuaregs, and to hold democratic elections.
Jakkie Cilliers, executive director of the Pretoria-based Institute for Security Studies (ISS), said the Bamako shooting showed the lack of legitimacy of Mali's current rulers.
"What now needs to happen is the political process, that includes the transformation of the local security forces," Cilliers said.
Traore has said he intends to hold elections by July 31.
Since last year's coup, elite paratroopers loyal to former President Toure had been largely sidelined and some were arrested following an attempted counter-coup in May.
Troops loyal to the March coup leader, U.S.-trained Captain Amadou Sanogo, put down the counter-coup bid in several days of fighting in the capital in which at least 27 people were killed.
Sanogo himself, who has nominally handed power over to an interim civilian government, has been accused by domestic and international critics of persistently meddling in state affairs. (Additional reporting by Adama Diarra in Bamako, Bate Felix, Richard Valdmanis and Pascal Fletcher in Dakar, Lamine Chikhi in Algiers, Adrian Croft in Brussels; Writing by Pascal Fletcher; Editing by Daniel Flynn and Roger Atwood)