A world that stopped spinning would not be anything like the Earth is now, and it certainly would not support life as we know it, so no, there would be no part of the Earth that we could live in. Bear in mind that this won't happen for many billions of years, so there could be quite a few other changes in living things, including us between now and then.
Your question is not totally unreasonable because there is something called the Earth-Moon tidal coupling which *is* indeed slowing the Earth's spin. It's believed that the length of the day was more like 18 hours one or two billion years ago.
For help figuring out the answer to your question, we can look at a neighbouring planet in the solar system that has very low-spin and also an atmosphere: Venus. Venus rotates slower than it revolves around the sun, so on the planet it would feel as though it had stopped spinning. Also there are very strong winds driven by the heat on one side and the cold on the other, and these winds blow the heat all around the planet, so Venus is not that much cooler on the unlit side, even though it is dark half the year.
But Earth's atmosphere is more than 90 times thinner than Venus's so the winds on Earth would not transport heat in the same way. A large heat difference would therefore develop between the sunlit side and the dark side of a stopped, or very slowly spinning Earth. I'm not sure how much, but we can guess that on the sunlit side the temperature would go up to an ocean boiling 150 degrees C, while the back side would be heated only by wind and ocean currents. If you consider that the center of the Antarctic ice cap is one place on Earth that is warmed only by wind transport of heat during its Winter season, and even with that warming it falls from a mean temperature of around -25C in summer to around -65C in winter. So one could imagine that it would get a quite a bit colder on the dark side if the Earth stopped spinning.
Another consideration is the greenhouse effect on Venus. If the Earth stopped spinning it would probably have this too, and it's pretty serious. As the ocean boils on the sunny side huge amounts of water vapour go into the atmosphere. Just as on Venus, at higher altitudes ultraviolet light would split the water molecules into Hydrogen and Oxygen. The hydrogen would escape to space because it is the lightest of all elements. The oxygen is very reactive so it would combine with other elements, most likely carbon, creating CO2. So Earth would start to become more like Venus with an extraordinarily thick carbon dioxide atmosphere and super hot greenhouse temperatures. Venus reaches 460C, but because Earth is farther away from the sun it probably would not get quite that hot.
I wouldn't want to live on such a planet and I'm not sure that anything *could* live on such a planet. But keep in mind, this won't happen for billions and billions of years.
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