Little Bear's Friend, written by Else Holmelund Minarik, illustrated by Maurice Sendak, Harper Collins, 1960.
I love Little Bear. And Emily. And Lucy. So it is probably no wonder that I named two characters in my first book 'Emily' and 'Lucie' -- even though I hadn't read a Little Bear book for about thirty years back then.
The illustrations are delightfully and perfectly vintage Sendak.