[git] Stashing only staged changes in git - is it possible?

With latest git you may use --patch option

git stash push --patch   # since 2.14.6

git stash save --patch   # for older git versions

And git will ask you for each change in your files to add or not into stash.
You just answer y or n

UPD
Alias for DOUBLE STASH:

git config --global alias.stash-staged '!bash -c "git stash --keep-index; git stash push -m "staged" --keep-index; git stash pop stash@{1}"'

Now you can stage your files and then run git stash-staged.
As result your staged files will be saved into stash.

If you do not want to keep staged files and want move them into stash. Then you can add another alias and run git move-staged:

git config --global alias.move-staged '!bash -c "git stash-staged;git commit -m "temp"; git stash; git reset --hard HEAD^; git stash pop"'