I ended up getting 5/64 and 7/64 drill bits. Now my problem is that the drill bits are too short (the 7/64 is 2.5 inches, and part of that goes in the chuck, whereas the screws are 3 inches) -- once I drill a pilot hole, I can get the screw in as far as the pilot hole goes, but no farther. I tried the lubricating the screw with soap as advised, but I still can't get it to go in any farther. The screws are going through the pine board easily, but once they hit the stud they start squeaking and finally they stop turning completely.
Strangely, I got the first screw I drove (back when I just put up the board like 2 weeks ago) to go in all the way, but the next 4 I tried all stopped moving with at least 1cm to go. I figured I'd take a couple of pictures in case that provides any useful information. This is what the board looks like now (most of the screws are just sticking out of the pine board, since I've only tried driving in about 4 different screws):
the surface I'm driving into appears to be a support for the roof of the building.
this is what my phillips bit looks like next to the deck screw I'm using:
the bit is getting kind of chewed up from getting banged against the screw head when the screw stops.
I guess my next step will be removing the board from the wall, and drilling pilot holes directly into the drywall, then putting the board back and driving the (now long enough) pilot holes.
It seems like it shouldn't be this complicated to do. Not sure why it's so hard. My leverage isn't great since I'm standing on a chair, but I'm applying considerable force while driving in the screws by pulling against the opposite side of the drywall as I drive in the screw.