A refigerator is a must have if you can possibly find room for it in your wet bar setup. Having to make regular trips to a fridge elsewhere to obtain fresh cold drinks and more ice is very much a second best solution.
Even if your wet bar is short of space, there are still options. For instance a compact under counter bar refrigerator will take less room than a normal fridge. Or you may be able to install a small refrigerator on a shelf, perhaps next to the shelf which a keg is on.
The ideal wet bar refrigerator will have a built in ice maker. To supply the icemaker however, you'll need to be able to tap a feed from the fridge to a nearby wall with a waterpipe. So you'll only be able to install one of these if the plumbing in your bar area supports it. The space under your bar sink might be an ideal spot. Some great bar refrigerators are featured below.