Green’s Dictionary of Slang

1. (orig. US surfing) a wood-panelled station wagon.

1963		Beach Boys ‘Surfer Girl’ 🎵 Surfer girl surfer girl / We could ride the surf together / While our love would grow / In my Woody I would take you everywhere I go.
1963		Jan & Dean ‘Surf City’ 🎵 I bought a ’30 Ford wagon and we call it a woodie (Surf City, here we come).
1973		J. Marks Mick Jagger 87: Woodies – those immaculate 1954 jobs by Ford: sturdy station wagons, armoured in wood.
1982		W. Safire What’s The Good Word? 54: My ’40 Ford Woody has a ding in the fender.

2. (N.Z.) a woodsman.

2003

McGill Reed Dict. of N.Z. Sl. 232: woody 1. A woodsman.

3. (N.Z.) a wood-burning stove.

2003

McGill Reed Dict. of N.Z. Sl. 232: woody [...] 2. A wood-burning stove.