Yes, drinks can have gluten. Be sure to check the label on anything that you consume to make sure it doesn't contain any gluten-containing ingredients. Sometimes a drink will list a flavor as one of its ingredients and not say if this ingredient contains any other ingredients, so always double-check the label on those sorts of things to make sure they don't contain gluten. For example, some sodas might say something like "sugar content: 12 grams (per bottle)." If "sugar" is listed as one of the ingredients without saying specifically what type or how much sugar it contains then there's no way for you to know if that specific brand uses sugar that has been refined using a process.