Could liquid methane support life

Not "life as we know it", such as life here on Earth. We can't speak with any precision about "life as we DON'T know it", other possibly exotic forms of life, but considering that in other environments that we once considered to be "extremely hostile to life" on Earth, life somehow manages to survive. For example, in ocean-bottom "fumaroles", places where volcanic activity is happening 15,000 feet or more deep in the ice-cold waters.

So, the completely accurate answer is, "Not that we know of, but anything is possible!"