Further rain warnings are in effect, with more than 250mm predicted to fall in some areas by Monday.
The Japan Meteorological Agency said the rainfall was "historic" and warned more rain was set to batter already saturated parts of the nation through Sunday.
Dozens of people have been killed and 2 million forced to flee their homes after record rains pounded southwest Japan, triggering widespread flooding and deadly landslides.
Television footage showed a massive rescue operation, with 2,310 rescued in the city by evening, according to NHK, while search and rescue teams continued to look for others.
Scores of patients, some still in their pyjamas, and nurses were rescued from the isolated Mabi Memorial Hospital in boats rowed by members of Japan's Self Defence Forces.
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The entire mission may last three to four days, as the boys are expected to come out gradually, officials said. He also taught the boys how to meditate and how to conserve as much energy as possible until they were found.
Public broadcaster NHK said flooding and landslides were hindering rescue efforts and repeatedly urged people not to lose hope.
According to their accounts, six people have been rescued from the landslide that struck houses in a residential part of the prefecture.
"I went to my father's family home but it was hopeless", one man said.
Chief Cabinet Secretary Yoshihide Suga said about 48,000 emergency personnel had been mobilized to rescue the stranded. "I was just able to escape, but I was terrified", 62-year-old Yuzo Hori told the Mainichi Shimbun daily in Hiroshima on Saturday.
Among the worst hit areas was Hiroshima prefecture, where some villages were hit by landslides and other were nearly entirely submerged, forcing desperate residents to take shelter on rooftops.
Although a weather front had settled between western and eastern Japan, there was a risk heavy rains would continue as warm air flowed towards the front, it added, with already-saturated areas facing more rain on Sunday.
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Rescue workers are now able to walk through a 1.5km (0.9 mile) stretch from the entrance to what's being called the third chamber. They have enlisted the help of bird-watchers, who are specialists in finding hidden holes, the AFP news agency reports.
The JMA also issued emergency warnings for exceptionally torrential rains in the prefectures of Ehime and Kochi on the western main island of Shikoku.
Okayama prefecture, south of Hiroshima, said in a statement that five people had died, seven were missing and 11 were injured, at least one of them seriously.
At least 4 million people have been advised to evacuate surrounding areas.
Two million people have been ordered to evacuate following flash floods and landslides in Japan, which has claimed at least 80 lives. Roads were also damaged and flooded everywhere and many railway sections remain disrupted.
Houses are partly submerged in water after heavy rainfall hammered southern Japan. "I've never seen this before", the 69-year-old oyster farmer told AFP.
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Passengers on the nearby boat can be heard screaming while others are seen wading towards the flaming ship with flotation devices. Chester Cooper , a member of parliament for Exumas and Ragged Island, identified the person who died as an American woman.