There is a workaround for stashing only selected files, but it is awkward:
https://community.atlassian.com/t5/Sourcetree-questions/Stashing-selected-files/qaq-p/137139

I frequently find myself wanting to commit portions of what I'm working on to a different branch. Being able to divide my uncommitted changes into different stashes so I can apply them to the desired branches would be very helpful. Example: I'm in the middle of working some one big feature when a simple bug is reported. I fix the bug in the development environment, test, and want to commit that one change to the released branch and push to staging/production.

Thank you for an awesome product.