PS: F1 are electronically assisted, not automatic (like the Boxter)
F1 gearboxes dont have the automatic facility because its not allowed in the technical regs. But adding it is just a matter of electronics I believe . Certainly the F1 box in my road car was developed by John Barnard for the 1990 season F1 season and is in effect an automated manual box with electronically controlled servos operating the clutch and gear selection.. I can use either flappy paddles or allow it to work in full auto. The Porsche and current Ferrari road car boxes are similar with double clutch and an extra layshaft but again electronics and servos allow you to use paddles or full auto.
Interesting quote from Patrick Head at Williams. " We ran the system on-track for the first time in 1990, but we didn’t race it. It was then part of the FW14 in ’91 and I think the system cost us that year’s championship because we had appalling reliability in the first four races. The problems were very easily solved and from mid-season we were very fast and reliable. All of our cars since have had semi-automatic gearboxes.
“Some people tell me that F1 would be better if the drivers still used stick shifts, but that’s a bit like saying, “isn’t it a pity we don’t still walk around in clogs!” There was a discussion in ’93 about banning semi-auto ’boxes because people like Alain Prost viewed changing gear with a lever as part of a driver’s skill-set.
“But we decided to keep semi-auto ’boxes because 95 percent of engine failures in the old days were caused by a driver changing down too early and over-revving the engine. The moment you have gear-shifting under electronic control, over-revved engines are out the door and life is much cheaper!”
Interestingly the Jag F type uses a more conventional automatic box re-designed to give better fuel economy by locking gears to prevent slip, and to give faster flappy paddle type changes. Effectively what seems to have happened is that modern electronics has allowed a "comoing together" of the old style manual and auto gearboxes so that both now offer the choice of manual or auto operation.