ThanksCustomerfor that additional information!
As you no doubt realize, alkaline batteries can cause BURNS if punctured. They can be serious. It is a relief that she did not swallow this, as this type of burn is much more serious in the esophagus than in the mouth.
You are doing the right thing to give her milk. You can give her a mixture of 50:50 water and milk - 1/2 cup NOW and then 2 tablespoons every 15 minutes. If you have lactose free milk that is even better, as the lactose can cause diarrhea tomorrow otherwise.
You may be able to give some aspirin (ASA, acetylsalicylic acid) as long as she has no history of kidney problems and is not on other medications. Please use Buffered Aspirin if you have it, and give it with a piece of bread or something low-fat to eat, not on an empty stomach.
Here are links that tell you about it, with precautions and dose:
You need to get your dog to the vet ASAP to get some sucralfate which is acts like a bandaid for damaged mucus membranes. The sucralfate can be given in pill or milkshake form and will help to protect the stomach and intestines.
When you get the dog to the vet, he or she will do a complete oral exam looking for burns. However, pain, inflammation, redness and swelling may not be visible for 2-4 hours after the battery was eaten. The full extent of the injury won't be known for 12 hours after exposure as the corrosive burns take that long to show up.
Treatment of burns in the mucosa should include antibiotics, pain medication as needed, gastrointestinal protectants (e.g. sucralfate), anti-inflammatories (corticosteroid use is controversial) and general supportive care.
In cases with severe oral burns or esophageal burns, your veterinarian may have to place a feeding tube into the dog's stomach so that food and water can be given to her through this tube while the mucosa heals. This can take weeks. There is a risk of a stricture or narrowing forming in the esophagus.
So, in summary, this may be a fairly serious depending on the extent of the burns. Please do get your dog to your vet promptly as soon as they open!
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Good luck, and please let me know what happens!