Some more detail would be welcome. We can't see your lock and we have absolutely no knowledge other that what you post.
Are we talking about a door handle?
Can you open the door and get to the other side?
Can you remove the handle the other side?
To use a screw extractor, you will need a hole in the end of the spindle - think for a moment how a screw extractor works.
If the spindle is steel, you MAY have some luck with a magnet but you would need something pretty powerful and small. I have frequently got honkingly powerful magnets out of old hard disk drives. these are ceramic and will easily snap into smaller pieces.
Consider replacing the spindle with one that is long enough not to disappear next time :-)
xxx
January 2017