I was thinking
Caddie Woodlawn, but that was written in 1935 about a girl living in the 19th century.
Anne of Green Gables was published in 1908, so that might be a bit late.
There's
Five Little Peppers and How They Grew, which was published in the late 1800s, I believe. Evidently there was a series of books published about this family, and it extended into the early 20th century. It's about five siblings growing up at this time, but the focus is on the eldest daughter, Polly.
I discovered this link, which lists 19th century girls' books, and it may have some useful titles. I don't know how available all of them are, but it might be worth looking. Looks like there are links to some of these.
https://www.readseries.com/
Oh, and if you want to go beyond books specifically about girls, there are children's books like
Black Beauty and
Beautiful Joe, which are still available in print and are intended to foster kindness to animals, which was an emerging concept in the late 19th century. The animal welfare movement actually informed the
child welfare movement in the late 19th century.
There are the
Little House books, of course. They were written later but the focus on the childhood and young adulthood of Laura Ingalls Wilder and her pioneer family in the 19th century. I loved those when I was a kid.
The
Awakening Land Trilogy was written in the 40s and 50s, but it's about a pioneer family in the late 18th and early 19th centuries.