Are new Braavosi key-holders created? If not, how is the issue of extinct family lines counteracted?

Good question!

So here’s what we know about keyholders:

“Archmaester Matthar’s The Origins of the Iron Bank and Braavos provides one of the more detailed accounts of the bank’s history and dealings, so far as they can be discovered; the bank is famous for its discretion and its secrecy. Matthar recounts that the founders of the Iron Bank numbered three-and-twenty; sixteen men and seven women, each of whom possessed a key to bank’s great subterranean vaults. Their descendants, whose numbers now exceed one thousand, are known as keyholders to this day, though the keys they display proudly on formal occasions are now entirely ceremonial. Certain of the founding families of Braavos have declined over the centuries, and a few have lost their wealth entirely, yet even the meanest still cling to their keys and the honors that go with them.

The Iron Bank is not ruled by the keyholders alone, however. Some of the wealthiest and most powerful families in Braavos today are of more recent vintage, yet the heads of these houses own shares in the bank, sit on its secret councils, and have a voice in selecting the men who lead it. In Braavos, as many an outsider has observed, golden coins count for more than iron keys.”

In the first place, the extinction of family lines doesn’t seem to be a problem, given their 43-fold growth. In the second place, to the extent that it’s a problem it’s been solved by allowing the wealthy and powerful to buy shares of the bank’s stock, and thus becoming voting members.