The former reporter, who was 90, died with his family around him on Friday lunchtime, a statement said.
His family described him as a “journalist to his bones” and he continued to file a column for the Yorkshire Post until as recently as last month.
Son John Ingham said: “To the wider world he is known as Margaret Thatcher’s chief press secretary, a formidable operator in the political and Whitehall jungles.
“But to me he was my dad – and a great dad at that. He was a fellow football fan and an adoring grandfather and great-grandfather. My family will miss him greatly.”
Born on June 21, 1932 and educated at Hebden Bridge Grammar School, Sir Bernard started his career in journalism aged 16 on his local paper in West Yorkshire.
He worked with the Guardian before becoming a Government press officer, but positioned himself as a bitter enemy of “spin”, criticising those who practised the “black art”.
Sir Bernard would handle the media as Lady Thatcher’s press secretary for all, but the first few months of her premiership.
He was knighted in her resignation honours and wrote a book, Kill The Messenger, about his life in No 10.
Sir Bernard was married to Nancy Ingham, a former policewoman, for 60 years. She died in 2017. He leaves a son, two grandchildren and a great-grandchild.
John Ingham thanked his father’s nursing home, Tupwood Gate in Caterham, Surrey, and his previous in-home carers for their “wonderful care and support”.