Mansfield 4000 series outdoor frost-free faucet leaking from inside the stem tube
It turns out that an o-ring (400/500 Stem End O-Ring in the diagram below) is completely absent. Since there is no leak when water is turned off, the washers should have been fine. When water is turned on, the stem end will be pushed against the thick copper part, thus closing the holes. With the o-ring missing, water drips through the seam, into the holes, and out from the handle thread.