Most do have a solid core.
As for life I don’t think anyone knows for sure if it’s impossible for life to arise there but most people would speculate that due to the extreme pressure, temperature, winds, and radiation it would be impossible. However, both life and the universe are constantly surprising us with what they are capable of so I can’t personally rule out life on gas giants completely.
Jupiter and Saturn are suspected to have solid or liquid cores. Even if they started off as only gas, the massive pressure at the core of the ball of gas that makes them up would cause the gasses to turn to liquid.
(Edit: tangentially related science – A lower atmospheric pressure also lowers the boiling point of a liquid, so for instance if you were to put a cup of water in a sealed pressure vessel and slowly remove the air towards a vacuum, the water would begin to boil off at a much lower temperature than 100 c. Conversely, if you raise the atmospheric pressure, the boiling point of a liquid rises, so under high pressure, it is possible to have water in liquid form above 100c, and there is a pressure “critical point” where the properties of a gas and a liquid are the same)
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