game of thrones - Are the Faceless Men and the Iron Bank connected?

It occurred to me that both the Iron Bank and the Faceless Men hail from the city of Braavos.

It is mentioned once or twice in the books that the Iron bank has funded wars for Kings and Queens in past struggles for power.

This is further recently acknowledged in Season 7 of the show with Cersei's hire of the Golden Company with money borrowed from the Iron Bank.

The assassins known as the Faceless Men have also been used for various political reasons. They are even considered for the assassination of Daenerys. One of the primary reasons they were not used was due to the cost.

"On Braavos there is a society called the Faceless Men," Grand Maester Pycelle offered.

"Do you have any idea how costly they are?" Littlefinger complained. "You could hire an army of common sellswords for half the price, and that's for a merchant. I don't dare think what they might ask for a princess."

_{A Game of Thrones - Eddard VIII}

With both of these things in mind, is there anything that suggests that the Iron Bank and the Faceless Men are connected in any way? I suspect that the Iron Bank may have gained their wealth through the earnings of the Faceless Men and enforce the debts they are owed through them.

Answer

There is no evidence that supports this theory, both in the books or in the show.

These two organizations without a doubt are based on the same Free City, but besides that, there is nothing that could suggest a direct involvement of one in the activities of the other.

The simple fact that their businesses, directly or indirectly, concern large sums of gold, is not a proof that they share common interests or that they are associates an any way: other banks are known to exist in the other Free Cities as well, and even some Great Houses in Westeros, mainly the Lannisters of course, but also the Tyrells, are known to be very wealty.
This does not mean that the Iron Bank, all the other banks, the Faceless Men, the Lannisters and the Tyrells all together are something of a single "cartel", just because they are rich.

Of course nothing prevents any of these parties to do business with the other (like in example, Lannisters borrowing money from the Iron Bank), so it is possible that the Iron Bank employs the service of the Faceless Men on the more desperate cases, but it seems that a usually it is sufficient for them to act against their debtors by supporting their enemies:

Tycho Nestoris had impressed him as cultured and courteous, but the Iron Bank of Braavos had a fearsome reputation when collecting debts. Each of the Nine Free Cities had its bank, and some had more than one, fighting over every coin like dogs over a bone, but the Iron Bank was richer and more powerful than all the rest combined. When princes defaulted on their debts to lesser banks, ruined bankers sold their wives and children into slavery and opened their own veins. When princes failed to repay the Iron Bank, new princes sprang up from nowhere and took their thrones.

_{A Dance with Dragons - Jon IX}

The World of Ice & Fire give more details about the birth of the Iron Bank, ruling out any involvement of the Faceless Men:

Braavos is also home to one of the most powerful banks in the world, whose roots stretch back to the beginnings of the city, when a few of the fugitives took to hiding such valuables as they had in an abandoned iron mine to keep them safe from thieves and pirates. As the city grew and prospered, the shafts and chambers of the mine began to fill. Rather than let their treasure sit idle in the earth, the wealthier Braavosi began to make loans to their less fortunate brethren.
Thus was born the Iron Bank of Braavos, whose renown (or infamy, to hear some tell it) now extends to every corner of the known world. Kings, princes, archons, triarchs, and merchants beyond count travel from the ends of the earth to seek loans from the heavily guarded vaults of the Iron Bank.
_{The World of Ice and Fire - The Free Cities: Braavos}

There are also details of the current management:

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. The bank's envoys cross the world, oft upon the bank's own ships, and merchants, lords, and even kings treat with them almost as equals.
_{The World of Ice and Fire - The Free Cities: Braavos}