The Lombardy region in the north of the country has been particularly hit hard with Brother Marco Bergamelli, one of the priests at the All Saints church in Bergamo, telling the that the church could not cope with the demand.
"Unfortunately, we don't know where to put them," Bergamelli said. "It takes time and the dead are many."
Crematoriums are said to be working 24 hours a day but still cannot keep up.
The latest figures released on Monday showed there had been 2,158 deaths and 27,980 known infections.
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The death toll in the country is the highest outside China.
Nurses in Italy have been taking to social media to reveal the state of patient care.
Some health care workers in Lombardy say the hospitals cannot cope with demand and they are running out of beds.