The rains, brought on by El Niño, are expected to continue for another two weeks, authorities said on Thursday.
President Pedro Pablo Kuczynski has for now ruled out declaring a state of emergency and said late on Wednesday that authorities are prepared to provide shelter and relief to those left homeless.
The downpours have caused floods and mudslides that have killed dozens of people. The city, where it nearly never rains, is home to 10 million people.
This dramatic video shows a woman pulling herself out of fast-moving muddy floodwaters in Punta Hermosa, Peru, on March 15. All the schools have suspended classes. They were feeding their pigs when the landslide seized control of them. They held on to each other's hands, but were eventually separated.
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The 32-year-old said she survived by grabbing on to tree branches and creating a makeshift bridge to clamber out of the sludge, reports Sky News. Upon reaching the land she collapsed, men at the location quickly helped her to an ambulance.
Mrs Chamorro Diaz was taken to hospital, where she is being treated for injuries but is in a stable condition, according to Peruvian media.
The Associated Press, Alliance France-Presse and United Press International reported on this story. George Grow was the editor.
"This emergency will pass, but we must be prudent", he wrote.