Do SMD mount bases exist?
There are breakout boards that allow you to solder the SMD part and then use it in something like a solderless breadboard or a lash-up by soldering to the large holes or adding headers to plug it in. You still have to solder the SMD part to the board once. The Chinese ones work fine. Photo from Aliexpress. You can probably find ones through the normal distribution channels as well.
This is a great way to play with new parts, test tiny things like 0402 LEDs and so on.
If you want to completely avoid soldering to the SMT part, there are sockets however they are huge and expensive and generally designed for applications such as programming SOT-23 MCUs. Not really a viable option.
Another option is to buy an evaluation board, if one is available from the chip manufacturer, or a 3rd party module such as those from Adafruit. This may be the most practical approach to deal with a BGA package in the early prototype stage. And sometimes the part you want to use is available in a more "friendly" package as an option, so you can use that and change to the unfriendly type for production.
Even if you may find compatible boards as shown in Spehro Pefhany's answer, it's very difficult to solder DFN or QFN packages manually. Even on an assembly line it's not easy. You need to apply solder paste in very tiny but exact amounts with very high precision. Then heat with a hot air solder gun. Succeeding or not is a question of luck.
The biggest problem is that you can't check the connections as the pins are hidden beneath the ic and not accessible to your multimeter probe. So debugging your circuit will be painfull as you will never know whether your schematic is wrong or the ic is not properly connected.
If possible use a package with visible pins (with the largest spread between pins available). If not possible, try to find a demo board (Development Board or Kit) for this ic. Sometimes they are expensive but will save you a lot of time.
The answer is, for some SMD packages, yes.
The component you are looking for is called a socket.
Digikey, for instance has an enormous number of sockets available, including those for BGA, PLCC and QFN packages. Many of these are designed to allow through-hole soldering to the board.