Mari-Simon Cronje, left, was at a children's birthday party when she suffered fatal injuries in the accident in a lake at the Princes sports club, near Feltham, west London, on Saturday.
A 22-year-old man, understood to be the driver of the motorboat which had been towing the girl, was arrested and bailed.
In a statement, the Cronje family said they were still battling to come to terms with the news.
Mari-Simon was taken to West Middlesex Hospital in Isleworth, where she died.
The Princes Club, which closed after the incident to allow investigations to be carried out, said in a statement: "Princes Sporting Club and everyone associated with it is devastated by the accident. At this terrible time our thoughts and deepest sympathies are with Mari-Simon Cronje's family and friends."