Was going to post this sooner, but have been busy... This may not be a special price, but you can get it now to see the Christmas greeting that was posted in the wee hours of Christmas Day, as well as the trailer. -The latter is available for free, FYI, but it looks like it will cost you, at least on Amazon, for the 2 min. "Happy Christmas" message if you want it alone. (Can't tell if it's the same on VUDU, since I've already pre-purchased the show there myself.)
...Have the 2019 New Year's special for Doctor Who available for $3.99 in SD, or $4.99 in HD. (On Amazon, click "More buying options" to see the SD offer. Clicking "Buy" attempts to purchase it immediately if you have a payment method on file, acting as if you clicked the 1-click buy button on a normal listing.)
I cannot tell if the price will be the same after it debuts online, but I doubt so, as Amazon at least kicked the price up about $10 more once this last season began.
So far, there's only two tidbits available on the page, but the title itself will be added to your collection as soon as it is released, presumably a bit after midnight PST on 1/1/2019.
(Note there is no Christmas Special this year... It's a New Year's Special!)
FWIW, Amazon calls this "Season 7", where VUDU calls it "Season 11".
Also FYI, it's not currently available on Google Play Movies & TV, but I expect it will at regular price once it's released.