Joel Lehman has created a Javascript-based Spotify playlist explorer.
The application does require you to login with your Spotify account, but then will display all of your playlists and allow you to export those as CSV files.
The resulting export is not a list of Spotify links, which is relatively easy to obtain, but rather CSV file containing the Track / Artist / Album / Date Added for all songs in any given playlist.
Lehman has a Github repository for the tool, but setting it up on your own server does require creating an app through Spotify’s developer program.