Yea Lead can be messy.
Slow speed on the drill.
Speed makes heat. Heat is the enemy, The tip of the drill generates most of the heat. The heat goes into the drill bit and the lead. Lead becomes highly ductile at relatively low temperature causing the lead turn from something like a rock into Play-Doh. Then, as it is being removed from the hole by the spirals, the lead cools and turns back int the rock. At which point, sooner or later, everything comes to a stop. The higher the spindle speeds, the sooner the stop.
After slowing down the drill, Peck at it..i.e. drill a bit, withdraw and drill a bit more...Like in the titanium drilling video.
Doesn't hurt to use some coolant. It improves tool life. Personally, I'm not sure the cost-benefit would pass.
In the event you choose to further explore this rabbit hole,
Check the same question here..
https://www.practicalmachinist.com/foru ... lead.86848