Your team schedules defects into your iteration backlog in the same way that it schedules user stories. During the sprint, your team spends time on the defect. But then you reach the end of the iteration and the defect is not completely fixed. What does your team then do with that defect? Do you split it in the same way you might split a user story?
If that defect is not completely resolved, then it is not considered done and should go back on the Product Backlog. The team gets no partial credit, but can bring the defect in to the next Sprint to complete it (if it is still important to the Product Owner). However, if the Product Owner sees true value in part of the resolution, then they have the right to split it up. But ideally, that is something that should have been done when first identifying the defect or at the beginning of the Sprint.