Most are staying with friends or family although almost 2,700 are being housed in hotels, schools, gyms and other centres laid on by local authorities, the Emilia Romagna region said.
Six months of rain fell in just 36 hours early last week, causing almost two dozen rivers to burst their banks, transforming streets into rivers and submerging huge tracts of farmland.
The clean-up has begun in some areas but others remain under water.
Efforts began Monday to restore an Internet connection to hospitals, public offices, schools and people in isolated areas.
The region said it had started to distribute 100 Starlink terminals from Elon Musk's SpaceX in the areas around Ravenna.
Aside from the loss of life, as yet unquantifiable damage has been caused to one of Italy's wealthiest regions.
Agricultural lobby Confagricoltura said at least 10 million fruit trees will have to be uprooted, and possibly as many as 40 million.
More than 600 roads remained closed on Monday, with the region estimating at the weekend that more than 620 million euros was needed to rebuild the water-logged road and transport network.
Prime Minister Giorgia Meloni returned early from the G7 summit in Japan to visit some of the worst hit areas on Sunday, and is expected to hold a cabinet meeting on Tuesday to unblock emergency funds for the region.
Several private groups have already pledged money to help Emilia Romagna get back on its feet, including Italian-French car giant Stellantis, which promised one million euros Monday, matching funds also pledged by Formula 1 and luxury carmaker Ferrari, which is based in nearby Maranello.
France's LVMH, the world's biggest luxury group which includes Italian brands Bulgari and Fendi, and Kering, home to Gucci, have also made unspecified donations.
The president of Emilia Romagna, Stefano Bonaccini, has drawn up a list of requests for government action, from compensation for flood damage to help for temporary agriculture workers and the suspension of tax and mortgage payments.
Longer term, he says the region needs a comprehensive plan "for the reconstruction, maintenance and safety of the territory", to address the flood risks.
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