Sure. Easy. Frst tell us which particular definition of "information" you are using, so we can select an appropriate example of it increasing.
Coded material fed to a computer or communications system. Specifically the information that controls the formation, development, and the 5000 or so chemical processes necessary for each cell to perform it's specialized function, and repair and reproduce itself.
Sorry, not usefual good enough.
You said "The barrier is that no new genetic information can be produced by mutation". In order to evaluate that claim we need a way to
measure the amount of information, so we can compare it before and after a mutation to see it it has increased or no;, or a way to
characterize information as "new" or "old" and then characterize the information present before and after a mutation to see if any of the information after the mutation is "new".
So, to be meaningful your defintion of information must include either a way to measure the quantity of information at any time and/or separate arbitrary information into "new" and "old" categories.
Of course, you wouldn't have made that claim without having made the appropriate measurements and/or determinations, or at least knowing a reference to someone who has ... would you?