There was a cat named “Strawberry” and the characters/kids in the book traveled to the Land of Now Here. I think at least part of the journey was completed in a hot air balloon, and there was a train.
Stump the Bookseller
New format, same book stumper solving.
Could be Beyond the Pawpaw Trees: The Story of Anna Lavinia by Palmer Brown.
Beyond the Pawpaw Trees is a tour through a land as strange and wonderful as Oz, filled with people as delightfully batty as any in Alice’s looking glass. It is a place to which you will want to return again and again, to read of Anna Lavinia’s adventures and to marvel over author and illustrator Palmer Brown’s intricate, sugar-spun drawings.
It all began on a lavender blue day. The kind of day, Anna Lavinia knows, when anything can happen. It was on such a day that Anna Lavinia’s father saw a double rainbow, and went chasing after it. And it is on such a day that she and her cat Strawberry set off on their journey beyond the walled garden where the Pawpaw trees grow, to a place where the buttercups bloom pink and the laws of gravity don’t always apply. Here she will test her mother’s advice, “Never believe what you see,” against her father’s wise words, “Believe only what you see.” And just maybe she will get to finally use her father’s mysterious silver key.
https://www.goodreads.com/book/show/1518739.Beyond_the_Pawpaw_Trees?from_search=true&search_version=service
Beyond the pawpaw trees by Palmer brown?
Beyond the Pawpaw Trees, and its sequel, The Silver Nutmeg (by Palmer Brown) are about a girl named Anna Lavinia, who has a cat named Strawberry.
Beyond the Pawpaw Tree by Palmer Brown?
More on Palmer Brown, if you like:
https://groups.google.com/forum/#!search/r.i.p.$20%22palmer$20brown%22/rec.arts.books.childrens/u_o6qduSduE/ThQTy6d0bE8J
He died on April 13th, 2012, five days after Maurice Sendak. He was 92.