Ok, here's my layers panel:
First, I added the Hue/Saturation layer. This is going to sound a little crazy, but go to every channel in the Hue/Sat dialog (Reds, Yellows, Greens, etc) and pull the Lightness slider all the way to +100 for each one. Don't do it on the Master channel, though. That will make the photo black-and-white, and lighten the background as much as possible.
It also makes the dog's tongue too light and desaturated, so mask strongly on the nose/mouth area, and lightly over the fur on the chest, to your taste. I hope you can see what I mean from the thumbnail in my screenshot.
Next, add a Levels layer. Pull the white slider in aggressively, to blow out as much background as possible. And fiddle with the middle slider a bit. I used 0.80 for the middle slider and 195 for the white slider.
Next, add a white Solid Color layer on top. Invert its mask to black, then start painting white where you want white. Use your best judgement for the shadow - I painted over the shadow with a 10% brush a bit, to make it a bit less strong. You'll know what you like best.
At this point I returned to the Background layer and made a new blank layer above it, for cloning. That was to get rid of some of the stones on the ground, that I couldn't paint out with white. Again, this will be to your taste.
Then return to the top layer. At this point, pour yourself a stiff drink, because you're going to be zooming WAY in and carefully masking around the fur for the bits of background that couldn't be made white by the Levels adjustment. I did a few easy parts, but as you can see, I left the hard parts.
This will take you a while. But it'll be worth it! I can't wait to see what you achieve.