Can I deep fry food in butter instead of vegetable oil?

You can certainly deep-fry foods in clarified butter (also known as ghee) and in lard. In fact, there are many foods that are traditionally fried in these fats. They both have very high smoke points and are excellent for making crisp fried foods.

For example, Puri, Indian fried breads, are deep-fried in ghee (clarified butter). And many Southern USA and many Mexican deep-fried foods are meant to be fried in lard, such as hand pies or sopes. In fact, if you watch the videos of Cowboy Kent Rollins, you'll see that while many of his recipes say "frying oil", what he actually uses is lard.

As for the flavor question: yes, using ghee or lard will affect the flavor of what you're frying, but in subtle ways. Both of these fats are mild-flavored (at least, high-quality lard is). Generally, the extra flavor you get from the butter or animal fat is considered desireable; they fell out of fashion in the use due to concerns about cholesterol, not taste. Only foods that are meant to have a very light, airy batter (like tempura) are unsuitable for frying in animal fat.

There are some other animal fats that can be used for frying and deep-frying, such as beef tallow, schmaltz, horse fat, or duck fat. These have a much stronger flavor that is recommended for specific foods (for example, there are many aficionados of duck fat french fries), but aren't a good general substitute for vegetable oil.

One other caution: if you switch to frying with animal fats, you need to make special provisions to dispose of the used fat. It can't be safely poured down the drain. This is actually true of all deep-frying oil, but animal fats are a greater problem: they may clog your pipes as well as hurting the sewer system.

ADD: per @wjandrea below, clarified butter, ghee, and brown butter have different flavors based on the amount they were cooked while clarifying, which will affect the flavor of any fried foods made with them.

No, you cannot deep-fry in butter. It simply can't handle the heat; it will brown and burn before you reach deep-frying temperatures.

In a comment you say that vegetable oils are unstable when heated, but it is in fact the opposite: butter is much more unstable when heated. Butter has a smoke point of 200-250F, around 120-150C. Many vegetable oils have smoking points of around 375F or 205C. Lard has a smoke point of around 370F, 188C, which makes it possible to deep-fry with in theory.

You can clarify butter and turn it into ghee, which has a very high smoke point, and you should be able to fry in it, though I havent done so.

As @ElectricToothpick said, the milk solids in butter will brown and burn, so that's not a good option. Since ghee has had the milk solids removed, that's not an issue.

Traditionally, rendered animal fats like lard were used for deep frying, and french fries were originally fried in beef tallow. McDonald's followed that tradition until health-conscious people made them quit. If you ever hear anyone talk about how McD's fries used to be better, its not nostalgia talking. It's the truth.

Frying in rendered animal fats gives you a crispier and more delicious final product. I should probably qualify that statement with "in my opinion," but I refuse to acknowledge people who prefer soggy fries. Someone else mentioned flavor but not texture, so I'll add that animal fats have a very different mouth feel and foods fried in them are going to lean more towards buttery and crisp than to greasy and hard in my experience.

Electronic Toothpick is correct about deep frying in butter. Lard, however, is perfectly acceptable for deep frying. French fries taste better fried in lard (imho). Solid fats in general are still used; especially in commercial establishments. The biggest drawback is waiting for the fat to liquefy and heat up to temperature compared to vegetable oils.

You can deep fry with any oil, it's all about taste and reusability of the oil. The higher smoke point doesn't just mean you can cook hotter, but generally the oil will last longer and can be reused more often.

I think taste is most important, it really depends on what you are cooking. Peanut oil is used often as it has the least noticeable taste. If I'm making tortilla chips I will use corn oil, shrimp is good with coconut oil. Eggs deep fried in lard is yummy, I had a relative that had a cast iron pan of lard on the stove at all times, cracked the eggs right into the oil, along with breaded summer squash slices. I've purchased frying oils that are a mixture of several oils, so mixing to get the flavor you want is always an option.

Ordinary butter is not good for deep frying as it has a low smoke point (~300F/~150C) and it consists of 17% water that would evaporate.