The blaze broke out in the eight-story structure on Rue Erlanger, near the Parc des Princes soccer stadium and the Roland Garros French Open tennis venue in the capital’s chic 16th district.
Paris prosecutor Remy Heitz told reporters that investigators suspected a "criminal" cause for the fire and that a woman was in custody.
French television showed video of flames in the windows of the building, which authorities said was built in the 1970s.
"It was a particularly violent fire," Captain Clement Cognon of the Paris fire service told BFM TV.
Emmanuel Gregoire, an official with the city government, said the victims were residents. The fire began on the upper floors and spread to other parts of the building, he added.
Paris Mayor Anne Hidalgo wrote on social network Twitter that the city was in mourning after the incident. It comes less than a month after a gas blast in central Paris killed four people.