A further 27 people are missing and as many as 41 children may be among the victims, Russian officials say.
The blaze started on an upper floor of the Winter Cherry complex while many of the victims were in cinema halls.
Video posted on social media showed people jumping from windows to escape the flames on Sunday.
Some 660 emergency personnel have been deployed in the rescue effort.
The cause of the blaze is not yet known but authorities have launched an investigation.
Kemerovo, a key coal-producing area, lies about 3,600km (2,200 miles) east of Moscow.
Austrian Chancellor Sebastian Kurz has offered his condolences to the families and friends of the victims, as did Latvian Foreign Minister Edgars Rinkēvičs.
As well as cinema screens, the complex, opened in 2013, includes restaurants, a sauna, a bowling alley and a petting zoo.
The fire is believed to have started at around 17:00 (10:00 GMT) in a part of the building that contains the entertainment complex, local media report.
“According to preliminary information, the roof collapsed in two cinemas,” Russia’s Investigative Committee said in a statement.
Yevgeny Dedyukhin, deputy head of the Kemerovo region emergency department, said the area of the fire was about 1,500 sq m.
“The shopping centre is a very complex construction,” he said. “There are a lot of combustible materials.”
Russian officials had initially given a figure of 64 people missing but later clarified that this included victims whose remains had not been identified.
At least nine of the bodies found so far are of children.
Andrei Mamchenkov, deputy head of Russia’s National Crisis Management Centre, said 41 children were not accounted for.
Emergency services finally reached a cinema hall on the third floor after being obstructed by smoke and the danger of collapsing masonry, an unnamed source told Russian news agency Interfax.
They found no bodies inside but fear people may have been buried under rubble.
Another source told the agency there was practically no chance of finding survivors. (Courtesy BBC)