Yes, it is possible but as you probably know there is so much to see that you would definately want to stay longer.
Arica to La Paz is around 8 hours if I am correct (by head, didn't travel it myself). The route is via Putre and the PN Lauca, a beautiful place and if you can, spend some time there, Lago Chungara is fantastic with the snow capped volcanoes reaching over 6000 meters!
La Paz to Uyuni can be done in a day as well, starting early. In Uyuni there are numerous options with tour operators, I went with tupiza tours (just google) but left from Tupiza, ending up in Uyuni. They probably leave the other way around as well. The thing is, they don't end in San Pedro if I am correct.
Uyuni to San Pedro can be done during a 3 day trip, or 4 if you prefer.
So of your 9 days of travel, it is one day to La Paz, one to Uyuni and 3 to San Pedro. Leaves you with 4 days more. I would spend one day in and around La Paz, one day Arica (or a daytour is you like) and a few days around San Pedro to see the El Tatio Geysir, Valley of the Moon or other wonders of nature. Then you still have one day to keep things flexible.
I am not totally sure, but flights back to santiago probably leave from Calama, a few hours from San Pedro.
Cheers and have fun!
Michael.