The strike on the base known as Camp Simba in Manda Bay is the latest by the group in Kenya since Nairobi sent troops across the border in 2011.
Kenya's army spokesman Colonel Paul Njuguna said in a statement that at 5:30am "an attempt was made to breach security at Manda Air Strip."
"The attempted breach was successfully repulsed. Four terrorists bodies have so far been found. The airstrip is safe. Arising from the unsuccessful breach a fire broke out affecting some of the fuel tanks located at the airstrip."
He said the fire had been brought under control "and standard security procedures are now on-going."
Lamu County Commissioner Irungu Macharia also confirmed there had been an attack and said the militants "have been repulsed."
"We are not sure if there are still remnants within," he said.
Al-Shabaab claimed responsibility for the attack in a statement, saying it had "successfully stormed the heavily fortified military base and have now taken effective control of part of the base."
The group said there had been both Kenyan and American casualties, however this could not be immediately verified.
Al-Shabaab said the attack was part of its "Al-Quds (Jerusalem) shall never be Judaized" campaign -- a term it first used during an attack on the upscale Dusit hotel complex in Nairobi in January last year that left 21 people dead.
The Somali jihadists have staged several large-scale attacks inside Kenya, in retaliation for Nairobi sending troops into Somalia as well as to target foreign interests.
Despite years of costly efforts to fight Al-Shabaab, the group on December 28 managed to detonate a vehicle packed with explosives in Mogadishu, killing 81 people.
The spate of attacks highlights the group's resilience and capacity to inflict mass casualties at home and in the region, despite losing control of major urban areas in Somalia.
The Lamu region, close to the Somali border, has been plagued by attacks from Al-Shabaab, with frequent strikes along the frontier notably targeting security forces with roadside bombs.
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