How can I git stash a specific file?

I usually add to index changes I don't want to stash and then stash with --keep-index option.

git add app/controllers/cart_controller.php
git stash --keep-index
git reset

The last step is optional, but usually, you want it. It removes changes from the index.

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Warning As noted in the comments, git stash --keep-index pushes everything onto the stash, both staged and unstaged. The --keep-index just leaves the index alone after the stash is done. This can cause merge conflicts when you later pop the stash.

EDIT: Since git 2.13, there is a command to save a specific path to the stash: git stash push <path>. For example:

git stash push -m welcome_cart app/views/cart/welcome.thtml

OLD ANSWER:

You can do that using git stash --patch (or git stash -p) -- you'll enter interactive mode where you'll be presented with each hunk that was changed. Use n to skip the files that you don't want to stash, y when you encounter the one that you want to stash, and q to quit and leave the remaining hunks unstashed. a will stash the shown hunk and the rest of the hunks in that file.

Not the most user-friendly approach, but it gets the work done if you really need it.