Baileys takes a very cautious approach when it comes to gluten: because whiskey (made from gluten grains) is used to make Baileys Original Irish Cream, the company won't say definitively whether or not the resulting liqueur is gluten-free.
"Due to the presence of gluten in the whiskey components before distillation we cannot absolutely guarantee that there are no effects to coeliacs. As we are not qualified to give medical advice persons requiring a gluten free diet should consult their medical adviser before consuming Baileys," the company says in its statement on allergens.
Now, despite what Baileys says in its statement, many (but not all) authorities on the gluten-free diet say that distilled alcohol sourced from the gluten grains wheat, barley and rye will be gluten-free, since the distillation process removes the gluten protein. (Some of us nonetheless have reactions to alcoholic beverages made from gluten grains for more details, see my article Is Alcohol Gluten-Free?)
That's still not clear (I wish I had a better answer!). The company keeps the details on the rest of the ingredients it uses to make Bailey's Irish Cream under close wraps, and it doesn't disclose if any of those ingredients (beyond the whiskey) actually contains gluten it simply recommends you consult your doctor.
Anecdotally, Baileys is a mixed bag some people report that it's fine, but others say they have had reactions to it. Based on all this, you'll need to use your own judgement on whether to consume Baileys Irish Cream when you're following the gluten-free diet.
There is a Baileys substitute that might work for you if you find you have problems with original Baileys: Carolans Irish Cream. In its Frequently Asked Questions, Carolans reports that it's both wheat- and gluten-free and "suitable for celiacs."
Note that the whiskey in Carolans is still made from distilled gluten grains (whiskey most commonly is made from barley). However, if you don't react to distilled gluten-grain-based alcohol, you should be fine, since Carolans has verified the rest of its Irish Cream ingredients to be gluten-free.
Another potential substitute for Irish cream is rum cream liqueur it doesn't have quite the same taste, but you can use it in the same way to spice up coffee drinks and to make martinis and other cocktails.
Three manufacturers RumChata, Cruzan and Sangster's make commercial rum creams. However, only one of these RumChata says its rum drink is gluten-free. Cruzan warns that the flavorings used in its rum cream may contain gluten, and I wasn't able to verify if Sangster's is considered gluten-free or not. So I'd stick with RumChata if you're looking for an Irish cream substitute that's made without gluten grains.