"The situation is nothing short of catastrophic. Ongoing fighting has killed or wounded hundreds of civilians who have only minimal access to medical care," said Director of Operations Pierre Kraehenbuehl.
A deadline for the rebels to surrender or face a final assault expired at 0630 GMT with no word from the Tigers.
"The [Liberation Tigers of Tamil Eelam] has not responded to the government's call to surrender, so we are keeping up our offensive to rescue the civilians," said army spokesman Brig Udaya Nanayakkara.
The Tamil Tigers said in a statement that more than 1,000 civilians had been killed and 2,300 injured during bombardments by government forces on Monday.
The rebel TamilNet website said the territory they still controlled was littered with bodies of civilians.
Video released by Tamil supporters shows mutilated bodies, but it is not clear when the recording was made.
They said it was filmed on Monday, but this cannot be verified.
It follows footage released by the government showing civilians fleeing the war zone.
One man who said he spoke for the Tigers, who gave his name as Thileepan, told the BBC a hospital, an orphanage and many houses had been hit.
He said people had been reduced to hiding under logs and trees and using makeshift bunkers dug into the sand.
The Sri Lankan military has denied shelling civilians inside the rebel-held area.
Brig Nanayakkara told the BBC that only small-arms had been used.
He said the Tigers were targeting civilians because they knew that if non-combatants left, the rebels would be "sitting ducks".
The army says three rebel suicide bombings had targeted fleeing civilians, killing 17.
Thousands of civilians are leaving the conflict zone
One Tamil man who had just left the conflict zone said the rebels tried to shoot anyone planning to escape.
Local newspapers are covered with pictures of large numbers of people leaving rebel territory, says the BBC's Charles Haviland in Colombo.
One calls the process a "human avalanche".
Reports suggest hospitals in the fighting zone were struggling to cope.
"We've had lots of dead bodies brought to the hospital, others are on the road," a health worker at Matalan hospital told the BBC World Service.
"Lots of the injured who are taken to the hospital die from blood loss - there is no blood-bank and a lot of people are anaemic because there is no food. People are suffering."
People escaped after troops broke through a fortification which had been blocking their advance into the Tigers' last stronghold, the army said on Monday.
TAMIL TIGER DEFENCES BREACHED BY SRI LANKA ARMY
12km-long defensive earthworks constructed by Tamil Tigers using mechanical diggers
Sri Lankan army uses explosives to destroy a 3km section
Gap allows thousands of refugees to flee Tiger-held territory
Aerial video showed thousands of people filing out of the combat zone. Tens of thousands remain in the area, which has seen heavy fighting for months.
The government says more than 49,000 have now left, more than 2,000 of them by sea on fishing boats.
They are on their way to government-run transit camps which Western media have no access to, it adds.
The pro-rebel TamilNet website said several hundred civilians were feared killed and injured after troops advanced into the zone.
Tamil protesters held angry demonstrations in Paris and London on Monday against the army operations.
About 180 people were arrested in Paris as the protest there turned violent.
Sri Lanka's defence ministry said on Tuesday troops were advancing into what was previously designated by the government as a safe zone.
It said another 3,000 civilians left the conflict zone on Tuesday.
Each side accuses the other of killing civilians in the long running civil conflict.
Foreign reporters are not allowed into the combat zone, making it impossible to independently verify the claims.
The Tigers are restricted to a 20 sq km (12.4 sq miles) coastal patch.
Gordon Weiss, the UN spokesman in Sri Lanka, said it was not known how many civilians remained there but that the UN had been working off a figure of some 150,000 to 200,000 people in recent months.
The government is not giving the ICRC access to the landward side of the zone.
So it can only evacuate people by sea, with two or three ships per week each carrying 400 or 500 of the sickest, oldest and most badly wounded people.
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