A Leisurely Life
Red dwarfs have one more trick up their sleeves. They're very very long lived. So long lived, in fact, that no red dwarf has ever died in our universe, because the universe isn't old enough yet!
The sun formed roughly 4.6 billion years, and will keep on burning just as it is today for another 5.4 billion years. By contrast, a red dwarf born at the same time would barely be a teenager. For example, Barnard's star - a friendly neighborhood red dwarf, just under 6 light years away - has a life expectancy of 2.5 trillion years. Over 200 times that of the sun. No one even knows if a planet can support life for that long!
While we know that life evolved here on Earth quite soon after our planet formed, we have no idea how long it would normally take for this to happen. If it's even remotely possible for the same thing to happen on a red dwarf planet, it would have literally all the time in the universe in which to do so!
Whether or not life exists is still an unanswered question, and one that we won't be able to answer until our telescopes are powerful enough to actually take a detailed look at planets around red dwarf stars. That said, with 13.5 billion planets to choose from, probability may well be on the side of the astrobiologists!