A new book on Donald Trump written by his niece sold nearly a million copies on the first day it went on sale in the United States, its publisher said Thursday.

Mary Trump's "Too Much and Never Enough: How My Family Created the World's Most Dangerous Man" is billed as the first unflattering portrayal of the US president by a family insider.

Mary, a psychologist whose father was Trump's oldest brother Fred, accuses the president of hubris and ignorance, and says he fits the clinical criteria for being a narcissist.

The 950,000 copies sold Tuesday, including pre-orders as well as audio and digital versions, "is a company record" for the Simon & Schuster publishing house, it said in a statement.

The White House has dismissed the memoir as a "book of falsehoods". 

The president's younger brother Robert Trump went to court to try to block publication, arguing that Mary was violating a non-disclosure agreement signed in 2001 after the settlement of her grandfather's estate, but to no avail.

Simon & Schuster has ordered more copies printed, which will bring the number to 1.15 million for the American market alone. The book also tops Amazon's sales charts in Canada and Australia.