I came across this idea in one of Hoops canoe videos. He starts with a thin, flexible piece of cutting board and attaches some closed cell foam to the bottom and a piece of cord. The foam looks like it helps grip the rock while cutting. The board can just float at waters edge getting cleaned and is away from camp in case of bears.
It's sort of hard to describe, but have a look at the video and just skip ahead to minute 8:00 to see the cutting board.
Hoop's cutting board If you are not familiar with Hoop, he runs a forum similar to this one specific to winter camping, and also does a lot of videos about his extended Canadian wilderness canoe trips.
If the plywood option works better for folks, that's great, but I'd suggest burning or packing out. My own preference would be to not find someone else's used cutting board in a campsite. Just my two cents.
Oh, and here's the
REI cutting board I think he starts with. Under 4 bucks!