Better answered questions: 2. Can life originate on gas giants? and 3. Can life survive on gas giants?
What I've seen of pop science, and my impression of Planetary Protection standards for spacecraft, is the assumption that because the answer to #2 seems to be "difficult", then the answer to #1 is "unlikely". But what if you drop living microbes through space and into the gas giant atmospheres?
That could be happening on a regular basis with Enceladus shown to drop water vapor from its geysers on to Saturn's atmosphere (blog post at link, pdf of article in the blog post). Enceladus probably has a near surface ocean feeding geysers through cracks in the ice. If it has life, then microbes are also being shot out as the geyser jets freeze.
The water vapor connection to Saturn doesn't necessarily establish that water ice particles will get there, or get there in the same short time frame (<2.5 months) as the vapor, but it makes it more plausible. And there's always the impact delivery of ice chunks from Enceladus to Saturn as a mechanism.
While microbes couldn't survive the deep level heat (which is what would stop life from originating on gas giants), if they can survive long enough to reproduce and randomly spread, then some descendant microbes can escape destruction. All they could need is some weather patterns keeping them out of the deep layers for weeks, unlike the millenia that would be needed for life to originate there.
Besides being interesting and a potential target for astrobiology, there are some policy implications to this. Crashing the Galileo orbiter into Jupiter may have been a mistake - better to have crashed it into Io, Callisto, or Almathea. The plan to eventually deorbit the Cassini mission into Saturn is also mistaken, when the dead-surface moon Mimas makes a better target.