Some Baptists fast. I've fasted. Just after I was saved I fasted for 10 straight days (it might have been more but I'm sure of 10) only drinking water. I'm not bragging and I wouldn't recommend that to anyone. In fact it was crazy. Not sure why I did it. It wasn't too long after I was saved so some of it was out of ignorance. But I was also going through a terrible time in my life. I dropped out of school, I was kicked out of the house by my parents, I was depressed and suffering from Pure OCD. The Pure OCD may have played into why I fasted so long. After the third day of fasting it actually got easier and even became hard to eat when I stop fasting. Kind of like how when you are so tired that you can't sleep. I'll tell you one thing, a lot of things happened in my life with God after that. It was bam, bam, bam. Maybe I should start fasting again. I won't be doing no 10 day fast though that's for sure.
One thing I noticed is that the early Christians fasted and prayed before they sent out missionaries (Acts 13:2,3). I know from the churches I've attended there was usually a big church supper before someone left for the mission field.