I’ve given her this background for specific reasons. Sally is a girl because women live about five years longer than men. She is 11 because, at that age, she has successfully navigated the diseases of childhood and her body has yet to endure the effects of ageing. She lives in Esher because it is one of the high life expectancy areas of one of the richest countries in the world.

Her parents are happy so they are less likely to divorce and cause life-threatening stress to Sally. They are wealthy because that means they can afford private medical care and Sally will not have to take the appalling risk of attending an NHS hospital. Finally, her grandparents lived a long time so the family has a history of longevity.

In short, Sally’s life prospects are optimum for a human child in 2005. According to current projections, she can expect to live well into her eighties. But it’s not going to be like that, because Sally is not going to die until 3194.

Hundreds of thousands of people die in the world every day, two-thirds of them from ageing. Is this just life, the way things must be, or is it a problem to be solved? If, as the western tradition teaches, every human life is valuable in and of itself, shouldn’t we be doing more to stop this appalling carnage? Or should we, as the eastern tradition teaches, accept it as the eternal becoming, samsara, or the veil of Maya, the illusion of existence?

Forget all that. This is what will happen to Sally. When she is at university in about 2013 she will hear news of an astonishing experiment performed on a mouse. A healthy two-year-old mouse will have been subjected to numerous protocols to suppress the division of cells and clean them of debris. In critical areas of cell depletion such as the brain and heart, stem cell therapy will have been used to rejuvenate the organs. Mitochondrial and chromosomal DNA will have been manipulated to stop damaging mutations.

The experiment began in 2010. The mouse is now five. Being an average mouse it should have died at three, almost certainly of cancer. At the genetic level, mice are surprisingly like humans.

Society is transformed by the news. In response to public demand, medical researchers are flooded with government money in the expectation that science is on the verge of delivering massive human life extension if not the holy grail itself — immortality. People begin to look after themselves fanatically, cutting out dangerous sports, smoking, excessive drinking, all sugar and red meat, and exercising daily. Everybody wants to live long enough to live for ever.

And sure enough in 2035, when Sally is 41 and beginning to feel the effects of ageing that humans have lamented throughout history, human rejuvenation becomes available. Privileged Sally is the first in the queue. Soon she looks and feels no more than 30 and, as the years pass, continued therapy ensures that she never ages. She has become, in fact, the first immortal.

But not, in the event, invulnerable. Accidents can still happen. Back in the early years of the third millennium an American biologist, Professor Steven Austad, studied death rates among 11-year-olds, the age at which disease is the least likely killer. On the basis of these figures, which included death by accidents of varying degrees of improbability, Austad calculated an “immortal” human was likely to live an average of 1,200 years.

And so, in 3194, blooming, youthful, beautiful, 1,200-year-old Sally is strolling along Esher High Street. A piano falls from a sixth-floor window and kills her. Sad, but never mind, she had a good innings.

All of this — well, not the piano — is exactly what Aubrey de Grey and an increasing number of scientists around the world expect to happen. De Grey is a brilliant, self-taught gerontologist at Cambridge. He is a 41-year-old cyclist with a 2ft beard, enormous whiskers and a rapid, high-pitched voice that on first contact is frankly terrifying.

He is in excellent health. He knows this because he has had one of the most rigorous medicals in the world at Kronos, an anti-ageing research institute in Phoenix, Arizona. But, in case it all goes wrong and he dies, he has arranged for his head to be frozen and stored by the Alcor cryonics facility in nearby Scottsdale. It will be revived when the technology becomes available so that Aubrey can go on talking.