The South African presidency was forced earlier this month to deny that the Nobel Prize winner was in a “vegetative state”, a claim made in court documents that also suggested his life support should be switched off.
However Zindzi Mandela said yesterday that her father’s energy had returned and he was responding to visitors.
“He’s making remarkable progress,” she said. “There was a time when we were very anxious. He continues to amaze us every day.
“I saw him yesterday afternoon with my mother and he was watching TV with his little headphones and gave us a huge smile.”
She added that the former president was responding to visitors with his eyes and sometimes his hands.
“You can see he is there in his eyes, the same energy and strength,” Ms Mandela said, adding she hoped he would be out of hospital “soon”.
Mr Mandela has been in a critical condition in a Pretoria hospital since June, after being admitted with a recurring lung infection. The world’s media had gathered outside the building doors, awaiting what was seen as imminent news of his death.
Ms Mandela’s description of his improving condition contradicts the claim that her father was in “vegetative state”, made by four members of his family as part of a legal dispute over the graves of three of his children.
Court documents dated 26 June said: “He is in a permanent vegetative state and is assisted in breathing by a life support machine. The Mandela family have been advised by the medical practitioners that his life support machine should be switched off.
“Rather than prolonging his suffering, the Mandela family is exploring this option as a very real probability.”
At the time, the South African presidency issued a statement that Mr Mandela’s condition at the time was “critical, but stable”.
Since then, various members of his family have said Mr Mandela was responding positively to treatment, prompting his wife, Graca Machel, to say last week that she was “less anxious” about the health of her husband.
Ms Mandela said the family was planning a special present for the former president on his 95th birthday.
She said yesterday: “Naturally it’s very difficult to come up with an ideal gift… So normally we just do huge, huge picture frames of all the family events and members of the family. We’ve got another huge collage to give him tomorrow.”
However, Ms Mandela said that a “true present” for him would be for the people around the world to each give 67 minutes of service to their communities, to match the 67 years the former leader gave to fighting for civil rights.
Mr Mandela, who spent 27 years in prison, led South Africa through a tense transition from apartheid to democracy and became president in 1994.
Hundreds of well-wishers have left messages of hope at his Johannesburg home and at the hospital in Pretoria.
And a group of young South African designers created a global poster project to raise money for a children’s hospital that will be named after Mr Mandela.