Oh, no. I've been there (cat). I have a regular mattress, but what worked for me was to call a local carpet cleaner that we've used before, with an excellent reputation and a schedule that let the guy come out immediately, before anything really dried. He used a powerful vacuum to suck the living heck out of the mattress, removing the pee, and then cleaned & deodorized the mattress. It turned out fine. I don't know what could have been done if the pee had dried first, though (and I didn't have time to read the info that Gumbeaux kindly posted for your type of mattress).
Another option, which would be a pain but which might help (even after cleaning), is to drag the mattress outdoors (yeah, loads of fun, I know). When my DS was a toddler, he peed all over a couch cushion. I blotted it as best I could, & then put the cushion into the blazing sunshine (not dappled or protected; I figured I wouldn't care about any bleaching as long as the cushion could be made to smell OK). It worked. The hot sun broke down the urine, I believe, and the cushion smelled fresh and clean after that.
Good luck. With your sweet dog, too.
If you're afraid of butter, use cream. ~~ Julia Child
As you cook, you enjoy omniscience about food that no amount of label reading can match. Having retaken control of the meal from the food scientists, you know exactly what is in it. (Unless you start w/cream of mushroom soup, in which case all bets are off.) To reclaim control over one's food, to take it back from industry & science, is no small thing; indeed, in our time, cooking from scratch qualifies as subversive. ~~ Michael Pollan