...Digging into the piece is only going to cause more damage. That'll need reworking.
Absolutely true.
But if the drill bit is in a place where you absolutely have to pin, with no space to redrill elsewhere, there is a way to remove it, but it'll be some work.
I've had several of these instances, mostly in model parts that I was in the process of pinning: regardless of the material (plastic, metal, resin), you can carefully cut away the material around the hole with the bit, to create a conical hole, with the drill bit in the middle. With a pair of needle nose pliers, you can now reach the drill bit and twist it out of the hole.
Often the breakage happens when the drill bit is not regularly lifted out of the hole during drilling, causing the cut away material to catch the drill bit, and the resulting torque can break it. And because of this, you can't pull the drill bit out straight, but instead, you'll need to reverse turn it out of the hole.
When the bit is out, you can continue deepening the drilled hole with a fresh drill bit, and when it's deep enough, put in the pin.
Then, when the pin is glued in, add some modeling putty into the conical hole, and presto; pinned and repaired...