Of course, lots of classic novels have been written this way. Dickens, for instance, wrote serials.
I’m writing a serial right now too, as it happens, only I’m cheating. I’m writing the whole thing as a single book, but building the climaxes so that I can divide it into four big parts. I call this cheating because it means that, unlike those other guys, I can go back and fix stuff.
If you are writing a serial in true serial fashion, I have two pieces of advice for you.
1. Keep your plots simple. Read Gentlemen of the Road or True Grit for examples. Make setting, humor, characters, and action the drivers of your story, and not byzantine plot.
2. Plan the whole thing in advance. Outline it in as much detail as you can, and especially know your major plot points cold. No room for exploratory writing here.
And if you get yourself into a corner, and have to cheat, by writing in a later fix for an earlier statement? Do it, and don’t apologize. That’s called “retconning“, and it has a long and glorious tradition, at least back to the Deuteronomists.