I vaguely remember this, Sid, from the time my kids were little. Did it come out in the 60s? Flat referring to the fact that he was two dimensional?
If I remember correctly a picture fell on him while he was in bed which squashed him flat. Although I did think that sticking him in an envelope with only an egg sandwich so he could visit relatives in America was a tad harsh.
Famous kids' book, Swale, by Jeff Brown. Stanley's shape enables him to have weird adventures. Kids love it. Funny and enthrals them. Check it out on Amazon