Yes, there is a bank and it is located in Protector’s Enclave. The bank is named Manycoins and can be found on your map. To access your map, press M on your keyboard. If you are having trouble finding it on your map, here is the location pointed out for you:
There are two personal bankers (Amber Lill and Kala Holt) and a guild banker (Fenton Udall) located in front of the Manycoins Bank. The guild banker in the middle is only usable if you are currently in a guild that has purchased at least the first Repository slot. The personal bankers are available to any character on your account and are located to the left and the right of the guild banker.
Each character on your account has his or her own personal bank account – that is to say, the personal bank isn’t shared between your characters. You have 16 starter slots available per character on your account. If you need additional slots, you’ll have to buy them using Zen. For 600 Zen, you’ll unlock an additional 16 slots. Keep in mind that these slots do not open account wide – for every character you’ll have to spend 600 Zen if you want to unlock 16 additional slots for his or her personal bank account. To buy these slots, just click the Buy Manycoins Bank Slots button located on the bottom left of the Bank screen that appears when you talk to either Personal Banker.