Thing about carbide tooling is it will not flex much without breaking. You MUST have work piece firmly mounted and use a drill setup with no run out. I only ever use them in my mill and with the small ones very carefully.

You might try spraying the bit with freon or soaking in dry ice to make it even more brittle then use a tiny punch to shatter it enough to remove it. Then use a slightly larger cobalt drill to clean everything up

Another method is to clamp the positive end of a battery charger on the work piece. Dampen the stuck bit and hole with a bit of salt water and then use the end of a small nail clamped to the positive terminal of the charger to keep touching the stuck piece of drill. This will slowly remove metal by electrolysis (reverse of plating).

Good luck