How can I drill a 2" deep hole, 1/16" in diameter?
A quick visit to a metalworking supplier website finds 3" long (overall length) ones with 1.7" cutting depth (presumably the flute depth, so for light-duty drilling if you "peck" (pull out to clear chips) when you get deep you could probably get it done. "Extended-Reach Drill Bits" A more extensive search might find a longer one. Buy them by the dozen, you probably will break some and they are usually cheaper that way.

A different metalworking supplier website has 6" and 12" OAL listed (under "aircraft drills" which the other site calls "Short-Flute Extended-Reach Drill Bits") but only 7/8" flutes, which would be a lot more pecking to get the chips out.