"keep on kicking" is a correct and usable phrase in written English. You can use it when you are encouraging someone to keep going and not give up. Example Sentence: "No matter how hard it gets, keep on kicking - you'll get there eventually!".
But even if the creatures did stay, did they hibernate or keep on kicking through the winter?
Keep on kicking ass, Jackie! .
Keep on kicking ass, Ronda. .
"We can put our heads in the sand and ignore that, and keep on kicking the can down the road, or we can come together as Sen. Coburn and Sen. Lieberman have with their particular proposal and try to do something about it," McConnell added.
Murakami's hand found a pillar and he struggled to hold on, kicking to keep up with the rising water even as it submerged him.
And Jonny Wilkinson kept on kicking goals from long distance.
You've got to love a story about a beautiful survivor who keeps on kicking.
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union