Is there a "Save Scratch File As ..."?

There is not. You may want to vote for the  IDEA-133703 Save Scratch Files to Disk feature request. In the meantime, as a workaround, you can use the "Copy" refactoring (F5) to create a copy which will be saved to the selected directory & name (it can be outside the project directory structure if desired).

Thanks! "f5" will work fine, and I'll vote up the "Save As.."

The "persistent" equivalent in IDEA 2017 are worksheets.
At least for Scala, you can create an "Scala Worksheet" that behaves like an Scratchpad, but can be saved anywhere in your project.

Much better than Copy is to right-click => Refactor => Move your file (or use the keyboard shortcut assigned on your computer). You have to select the correct location and when you're done the file is there! Verified with PhpStorm 2017.3.

Note Mac users have right-click 'Reveal In Finder' too.