I'm with those who say each character's dialogue should be on a new paragraph.
I also think breaking up a character's dialogue into multiple paragraphs is fine. Generally in my experience when this is done writers will not put a closing quote mark, for example:
'A character may speak for a long time, perhaps telling a story or recounting some important lesson, or attempting to inspire their armies to war.
'If a paragraph break seems reasonable the first paragraph won't have a closing quote mark, but the new paragraph will have an opening quote mark. This ensures the reader knows it's still the same speaker.'
Make sense?
Ultimately though, I think as long as the reader isn't confused, you're fine. I, for example, try to avoid dialogue tags as much as I can when dealing with a short snappy exchange. It's a technique I first experienced reading Terry Pratchett, and I definitely prefer it, probably because of my background in film where obviously screenplays don't have dialogue tags.