Nope. The history goes a littel something like this:
1st Doctor - No sonic screwdriver.
2nd Doctor - The sonic screwdriver made its first appearance in the serial Fury from the Deep, written by Victor Pemberton. It was a simple metal tube body pen light.
3rd Doctor - The BBC bought up a load of props boxes from the production company that had produced, among other things, the Thunderbirds. In one of the boxes was a hand tool, screen used in Thunderbirds Go! that the production team painted one part yellow and added a radar dish looking thing to the top part. Jon Pertwee carried that, you can see it well in The Sea Devils. As time passed the prop was cleaned up, polished and a new head was put on. That version has a couple nice close ups in Frontier in Space.
4th Dcotor - Tom Baker's version was the same one as Pertwee's but got more and more beat up over time. There's some talk that the original vanished by now and had been replaced (possibly more than once) by new ones made by independant prop masters on the BBC's payroll.
5th Doctor - The one he carried is obviously not the same one as before. It has different proportions even though the description is the same. A PVC dummy was made for the scene in The Visitation where it is destroyed.
6th and 7th Doctors - No sonic screwdriver but he did have a sonic lance.
8th Doctor - Iguana Props is asked to bring back the sonic screwdriver after a considerable absence (from the early 1980's to 1996) and makes a new version with a telescoping section, a brass band around the handle and a pen cap shaped head instead of a bullet shaped head. The film used on hero model (which Sylvester McCoy broke) and a few wood stunt models. None of the versions from the 1960's-1980's showed up in the 1996 film, the props had been custom fabricated for the film.
9th and 10th Doctor - Totally new prop.