I've noticed that our Black and Decker battery-powered chainsaw seems to
drink blade-lubricating oil. A tankful (about 50 ml at a very rough guess)
seems to last only a minute or so of operation, with a lot of oil being
splattered off the blade onto anything in range (gardening clothes, tree
being pruned etc). When I came to it today, it was sitting in a small puddle
of oil.

Is there any way with chainsaws of regulating how quickly the oil flows out?
The instruction manual doesn't actually label the port through which oil is
released onto the chain and I can't see an obvious hole. With the chain and
chain guard removed for cleaning, and the tank full, I can't see any oil
dripping out, so it's evidently happening too slowly to see a drip forming
when the motor isn't running.

The model is B&D GKC3630L20 and the manual and sale literature just says
"simple, leak-free oil system with automatic chain oiling" without anything
about what to do if the automatic process empties a tank in a minute or so.