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Surely a "local private wiki ... not web based ... on a desktop application" is not really a "wiki" at all, but rather a "static site generator" with a built-in "search". If that's what you want, there's a Python app called Pelican. Writing such an app from scratch isn't really a beginners project.
As a starting point, take a look at the Python-Markdown library. It's available from the Pypi repository, so is easy to install with pip / pipenv / ...
NOTE:
The number of mentions on this list indicates mentions on common posts plus user suggested alternatives.
Hence, a higher number means a more popular project.