For various reasons I can confirm that even a 9.5 client will still "work" with an 11.5.9 database.
It is not recommended, but these things happen when people drag old modules from the depths of SVN or wherever they hid it.
So, yes, the 9.7 client will probably work on 11.5.9 but it is unsupported, and you won't be able to use any new functionality or features.
You may also be sitting on a bunch of security issues that have been addressed by later client versions.
I would recommend updating the client to 11.5 if it is at all possible.
There are a large number of changes and improvements from 9.7 to 11.5, make sure to read the Db2 docs to check for any impact to your applications as well.