Videos circulating on social media showed flames engulfing a five-story building in al-Kut overnight as firefighters tried to contain the blaze.
Reuters could not independently verify the videos.
"We have compiled a list of 59 victims whose identities have been confirmed, but one body was so badly burned that it has been extremely difficult to identify," a city health official told Reuters.
"We have more bodies that have not been recovered still under fire debris," city official Ali al-Mayahi said.
The cause of the fire was not immediately known, but the province's governor said initial results from an investigation would be announced within 48 hours, the state news agency (INA)reported.
"We have filed lawsuits against the owner of the building and the mall," INA quoted the governor as saying.