I have always understood this to be a Vermont/New England term, deriving from "driving like a banshee".
Verb - 1.The act of driving very fast on a winding country road, preferably a dirt road. 2. The act of purposefully driving off the roadway and into a field, especially after much rain, where through decent pedal-work the vehicle can be made to spin, providing a fun ride and tearing hell out of the field.
"We were banchin' up Old Stage Road going wicked fast."
"Jeff's lawn was soaked after all that rain, so I went banchin' on it."