After all, this is a ladies’ day, organised by the British sports-car maker and Formula One competitor to take the testosterone out of supercars and demonstrate how women drivers can handle them, too.

There’s even a professional woman racing instructor with us.

The carmaker runs regular track days for customers, and has discovered that wives are as passionate as their spouses about the boys’ toys. Women, McLaren believes, can feel just as confident behind the wheel as their man, and is considering hosting more events internationally, including in Asia.

Not a single woman in our group of almost 20 emerged from the supercars without wobbly legs and adrenalin pumping hard. It was a truly exhilarating experience.

My first lap in the McLaren is tentative, as I learn where to hang wide before aiming at the apex of the bend, then put my foot down again, as Sutton, surprisingly calm, talks me around the circuit. Advanced driving lessons years ago are quickly recalled and I glance at the speedometer – 105km/h before the chicane. Surely I can do better.

With each lap the speed increases, the time drops, and my confidence builds as I realise the car is not going to send me skidding onto the grass every time I put my foot to the floor. The air brakes create more downforce at speed so the wheels are glued to the road when cornering.

Switching up through sport mode to track mode, the suspension stiffens, and the car’s traction control is phenomenal when the brakes are applied. The setting, I am told, optimises performance and on the straight. It sure does, as we accelerate from 48km/h to 153km/h in three seconds. This is fun!

It’s a revelation how light and responsive the steering is. The car feels so stable and is far easier to drive than I expected, even though I’ve never used paddle-shift gears before. The drive settings can be switched between automatic and manual gear-changing mode. I am amazed I can even talk with Sutton through the many laps of the circuit.

The top speed of these cars is 333km/h so my 153km/h would not have impressed Clarkson, especially as he drove the even more exclusive McLaren P1 in the “Holy Trinity” race against a Ferrari and Porsche in the debut of his new Amazon Prime series, The Grand Tour.

My feeble excuse was a couple of temporary chicanes. Nevertheless, I’m pleased to cut my lap time by 30 seconds, to 2 minutes and 30 seconds, by the end, even though another woman driver breaks the two-minute barrier. But then she’s the motoring editor for a luxury magazine and learned motor racing at the prestigious Caterham Academy, so I don’t think she counts.

The hot lap with Sutton at the wheel is something else. It has me hurtling back to Earth as the G-force hits my chest when we accelerate away from the pit lane, then slam on the brakes for the first turn, in less than three seconds. I suddenly regret having a cup of coffee before getting back into the car.

In Sutton’s hands, the demon in the McLaren is unleashed as we blast from 8km/h to 160km/h in just 5.7 seconds. The performance of the 650S pin-up is literally breathtaking. When we stop, I think my heart does, too, having exhausted every drop of adrenalin in my body.

After the track laps, we swap the 650S for another McLaren sports car, the 570GT. We take this on a gentle cruise around the Goodwood Estate, up the legendary hill climbed by so many racing legends including Bruce McLaren himself, before heading out on the roads. I’m told McLaren cars are just as comfortable for commuting, doing the school run (provided you only have one child) or a trip to the supermarket. Fortunately, my local supermarket has very wide parking spaces because I wouldn’t trust any other car parking near this.

Nevertheless, more stowage under the bonnet and behind the seats would handle quite a few shopping bags, or small suitcases if planning a weekend away, but there’s no space for the dog.

There are several women owners present for ladies’ day and I, for one, would love to return, next time for a spin in the McLaren P1. It’s the haute couture frock of the supercar world, rather than the luxury ready-to-wear 650S and 570S.

The fashion metaphor is not entirely inappropriate; there are other signs that the world of supercars is not purely a male domain. Some of McLaren’s design team members are women in what is traditionally a masculine arena.

Donna Falconer heads project strategy for the carmaker, having studied design and engineering at university. She’s the one with the crystal ball, predicting market trends and where McLaren fits into the picture before putting together the design brief.

“The key thing for me is our cars must be exciting to drive, while inspiring confidence at the same time,” Falconer says. The first time she drove a McLaren, she says, “I thought I was getting into a spaceship”. She says she soon found them easy to drive, and very flattering to the driver. “They make everyone feel like a racing driver.”

Joanna Lewis, the senior colour and materials designer, has a fashion background. She studied at Central Saint Martins in London, then worked for Stella McCartney and her husband Alasdair Willis for his furniture design business, Established & Sons, before joining the car industry at Jaguar Land Rover and then Lotus.

Car design may be a man’s territory, but she says her role is dominated by women. “It is a fact that women have better colour vision than men, so maybe that’s why,” Lewis says.

More women are coming to the fore; there are a number at management level at McLaren and a woman runs the assembly line in the factory. “We have a lot of female customers, so as we design we need to see the product from both sides [of the gender gap],” Lewis explains.

Amanda McLaren is the ambassador for the brand. As founder of one of the most successful Formula One teams in history, it was her late father Bruce McLaren’s dream to develop a road car from a racing car until his life came to an end tragically in a testing accident at Goodwood in 1970. Amanda was just four years old. Others carried on his pioneering work in what is now the sister company to the racing team, McLaren Automotive.

“Nothing drives like a McLaren because nothing has been designed like a McLaren,” Amanda says. “From the M6GT, designed and engineered by my father, to the new Sports Series, we’ve been transferring racing technology to the road.”

She has never been tempted into racing herself, however. “Whether by nature or nurture, I really enjoy cars and driving. I enjoy the speed and the performance of the cars, but my preference is testing and development.”

As for what her father might have thought about this ladies’ day, she says: “He would have been delighted and very encouraging.” As for his female guests: can we do this again please?