* Al Qaeda-linked rebels extend deadline for officials
* Shabaab have demanded Kenya frees Muslim prisoners
* Nairobi has said cannot negotiate with "terrorists"
By Feisal Omar
MOGADISHU, Feb 15 (Reuters) - Somali militants linked to al Qaeda said they executed a Kenyan army private at midnight on Thursday after the expiry of an ultimatum for the release of Muslims held by Kenya on terror charges.
The al Shabaab militants, who want to impose a strict version of sharia, or Islamic law, also said they were extending the execution deadline by 72 hours for five other hostages.
The insurgents on Jan. 23 had given Kenya a three-week deadline to free fighters held by the east African country, but Kenya refused to negotiate and said it would not meet the demand.
Prior to the ultimatum, the group had killed French hostage Denis Allex to avenge France's persecution of Muslims and its military operations against Islamists, including in Mali.
The Kenyan defence forces have said none of its soldiers was believed held by al Shabaab.
Security analysts describe al Shabaab as a declining force that no longer controls much territory in Somalia and is not a threat to the Somali government or African Union forces there, but is still capable of mounting sporadic attacks
Nairobi sent its troops across the border to pursue the rebels in October 2011 after the insurgents kidnapped Western tourists and Kenyan security forces on Kenyan soil.
The rebels said there was still a chance of securing the release of the five prisoners, who include two government workers.
"Despite the Mujahideen opening the doors of negotiations and giving them ample time, the Kenyan government has been wilfully negligent and has failed to take adequate steps to secure the release of its citizens," the rebels said in a statement, in which they gave the executed soldier's name.
Kenya holds several suspects for alleged links with al Shabaab and extradited others to Uganda suspected of involvement in a deadly 2010 suicide bombing in Kampala, claimed by al Shabaab, that killed 76 people.
The rebels said they would give more time to the remaining hostages in response to their families' pleas.
"In 72 hours, the prisoners will either lose their lives because of their government's betrayal or celebrate their government's assistance and rejoice in their freedom," the militants said.
The militants released a video titled "KENYA POWS: FINAL MESSAGE", which showed Mule Yesse Edward, a local administrator, and Fredrick Irungu, who worked for the Kenyan ministry of immigration.
Both were captured last January when the militants crossed the border into Kenya and attacked a police post in Wajir county, killing several police officers.
The video also contained a still photograph of four unidentified prisoners whom it said were also Kenyans. (Additional reporting by Abdi Sheikh; Writing by James Macharia; Editing by Michael Roddy)