Is there a standalone free offline version of Minecraft?

Since we clarified you're not looking for a pirated version :-) There are three choices:

• Minecraft Classic, which is very old no longer available, lacks e.g. redstone and zombies, but is still great if you like simply to sandbox.
• The Demo, which is not up-to-date but features redstone and is in survival mode (health bar, swords, spiders...). Any world you create there can however only be modified for 90 minutes
• The full version (you have to buy it in this case), have it downloaded by someone else and have them snail-mail you the whole folder %appdata%\.minecraft\.

edit As John commented, the internet-heavy part is the download, so for the third solution you can ask said person use their login, send you their .minecraft (excluding the file lastlogin to keep it legal, and the saves folder isn't necessary either) and then connect to the internet once to login with your own account. If there has been an update meanwhile, you can choose "update later". After the first login, it will work without internet connection

To avoid snail mailing, your friend can also upload the folder to something like dropbox and you can use a download manager. It's only about 50 MB after all.

Or if you do not want to use the Internet to log in, you can use the following command (that goes in your run box) to start Minecraft without logging in (assuming you have the .minecraft files in %APPDATA%

java -Xms512m -Xmx1024m -cp "%APPDATA%\.minecraft\bin\*" -Djava.library.path="%APPDATA%\.minecraft\bin\natives" net.minecraft.client.Minecraft` yourusername

No, there is not.

However, as you pointed out, the paid version can be played offline indefinitely after you've downloaded it. I'm not sure what difference a free version would make in that regard, as you'd still have to download it.

Actually there is, and its called Minecraft Classic. You can find a download link to it currently on the main minecraft.net website on the right hand side. It is a very out dated version of the game however and is pretty much akin to the Creative mode that minecraft has now.