Yes, milk is gluten-free.
All types of plain cow's milk are naturally free of gluten. However, some dairy products are not gluten-free. Once flavorings or other ingredients have been added to milk it may no longer be gluten-free, so it's important to read the label to see if the product contains gluten or not. Additionally, avoid malted milk drinks as they are not gluten-free. The word "malt" on a food label usually indicates they are made with barley and therefore contain gluten.
It’s also important to note that while milk is gluten-free, for those newly diagnosed with celiac disease, secondary lactose intolerance is common due to the loss of lactase, an enzyme that digests milk sugar along the lining of the small intestine. The damage that gluten causes in the small intestine is the main factor in the lack of lactase for people with celiac disease. While following a strict gluten-free diet, the gut is able to heal, making lactose intolerance temporary in most celiac disease patients.
Learn more about celiac disease and lactose intolerance here.