I would expect poor results with a drill - as the cutting edge comes perpendicular to the fibers it will no longer be cutting them but prying them up in a way that they are very strong compared to the matrix/resin ability to hold them. Once loose they are pushed clear leaving shards around the hole.
In wood and thin sheetmetal the same prying action can take place and is far worse on the far side as the pressure of the advancing drill tend to delaminate it even before the cutting edge can get there making a huge mess.
Of course this is what Composite Pro said - just repeated for emphasis.
So, with that in mind, I would look at diamond dust coated grinders to at least start on the the initial layers - like a countersink or a dent. This way the upper layers of the composite, now out of the way with some small clearance and not seeing peeling forces, can act as a local backer for the lower layers.
Same relief is done on hand planing end grain in wood to keep a chunk from splintering - put a chamfer where the cutting edge leaves the material so that there is support right there.
Here's an example of a backer for wood cutting:
But it would also work to have a chamfer where the backer is that is deeper than the cut so the edge of the board never sees a cutting load - the required depth of the chamfer will depend on how much support is needed. I tried to find a better video or picture but everyone is concerned about tearout from the chamfer on the endgrain - which, if you start with the diamond dust coated grinder you won't have.