"Unfortunately we have more than 35 (dead), and we still have houses to check," Buenos Aires Gov. Daniel Scioli told Argentine radio.
Scioli spoke of a "tragedy that is without precedent and will have extremely serious consequences," he said.
Several hours after the rain stopped, many streets remained under water.
La Plata, a city of 650,000 about 50km south of Buenos Aires, is the capital of the Buenos Aires province, which surrounds the Argentine capital.
The heavy rain that began falling late on Tuesday (local time) became "a fatal trap," said Scioli.
"Our greatest concern is to save as many lives as possible," he stressed.
About 40cm - or more than four times the average rainfall for the whole of April - fell on the city in the course of a few hours, forcing 3000 people to flee their homes. Many climbed rooftops and trees to escape the floods.
In the early hours of Tuesday, another storm with 20cm of rainfall flooded several neighborhoods of the city of Buenos Aires and its suburbs, killing eight people.
