* France has 6-month mandate to restore security
* Looting and killings rampant
* France and U.N. condemn lynching in capital (Adds U.N. quotes from Bangui)
By Maya Nikolaeva and Media Coulibaly
PARIS/BANGUI, Feb 6 (Reuters) - French Defence Minister Jean-Yves Le Drian said on Thursday the United Nations would probably have to renew a mandate for French troops to restore order in Central African Republic when it expires in May because of continued violence there.
France deployed 1,600 troops to its former colony in December to stem violence between Christian militias and largely Muslim Seleka rebels who ousted President Francois Bozize last March.
Asked during a radio interview whether France's U.N. mandate, granted on Dec. 5, would have to be renewed, Le Drian said: "Probably."
He said that the French mission had helped ease tensions in Central African Republic, a landlocked nation of 4.5 million people, and that the security was improving day by day.
"The situation in Bangui has more or less stabilised," Le Drian told RTL radio. "In the rest of the country it is much more complicated."
France had hoped to quickly hand sole responsibility for security to the African Union peacekeeping force MISCA, which has deployed over 5,300 troops to Central African Republic, including soldiers from Rwanda, Burundi, Chad and Cameroon.
Close to one million people, or a quarter of the population, have been displaced by clashes that began when Seleka rebels embarked on a looting and killing spree after seizing power.
Christian self-defence groups known as "anti-balaka" (anti-machete) took up arms against them. The United Nations estimates that more than 2,000 people have been killed in the violence.
In a sign of tensions running high in the capital, government soldiers on Wednesday lynched a man they suspected of having been a rebel.
The lynching took place minutes after interim President Catherine Samba-Panza called for reconciliation in an address to troops, and was condemned by France and the United Nations. Both called for an investigation and appropriate sanctions.
"The incidents yesterday are unacceptable. They must be investigated and punished," General Babacar Gaye, U.N. envoy in the country, told a news conference on Thursday.
Samba-Panza said in her speech on Wednesday that she hoped to restore order within a month, though Human Rights Watch (HRW) has warned Seleka fighters who fled the capital have regrouped in the northeast and begun attacks on civilians. (Additional reporting by Chine Labbe and John Irish in Paris; Writing by Bate Felix; Editing by Raissa Kasolowsky)