I’m not sure what settings you’re using in Unity when importing. Are you using the latest version of Unity? In previous versions there was a vertex limit and when you went past that on a single piece of mesh it would cut it up for you. Also if you have any kind of modifiers on your character in Blender I would apply them and export.
You could troubleshoot it a bit more to see if it’s the rig of your character or just the character in general. Export the character solely as a mesh to see if that’s the issue. If you’re using a custom rig setup like Rigify or something along those lines it may cause issues in Unity as it wouldn’t recognize a lot of it.