This feature has been implemented and will be available in Sourcetree 3.4.24 onwards.

Neither of these solutions in the comments work if you want to stash multiple files that have a space in the directory name... Does anyone have a solution? putting "$FILE" in quotes works for one file but not multiple.

698877135425 is this still being worked? I would use this frequently if it were available in SourceTree! 

We are working on this feature.

I created a batch script to stash the selected files only. With this solution you can also provide a message for the stash entry.

@echo off

REM Create a temporary file for entering the stash message
set "input_file=stash_message.txt"

REM Open the file with Notepad
start notepad "%input_file%"

REM Wait until the file is saved and closed
powershell -Command "while ((Get-Process notepad -ErrorAction SilentlyContinue) -ne $null) { Start-Sleep -Milliseconds 500 }"

REM Read the content of the file into a variable
for /f "usebackq delims=" %%a in ("%input_file%") do (
    set "stash_message=%%a"
)

REM Delete the temporary file
del "%input_file%"

REM Initialize variable for the selected files
set "selected_files="

REM Concatenate every argument using a space as delimiter
:concatenate
if "%~1" neq "" (
    if defined selected_files (
        set "selected_files=%selected_files% %~1"
    ) else (
        set "selected_files=%~1"
    )
    shift
    goto :concatenate
)

REM Stash the selected files with the provided stash message
git stash push %selected_files% -m "%stash_message%"

After you created such a .bat file, add it as a custom action in the following way:
1. Give an arbitrary name for the custom action in the "Menu caption" field.
2. Only "Run command silently" has to be checked.
3. Browse for the created .bat file in the "Script to run" section.
4. Set the "Parameters" to $FILE.

After you selected all the files that you want to stash and clicked the custom action, a text file will open where you can provide the message. Write the message in this file, save it and then close it. The selected files will be stashed with the provided message.

Make sure you actually browse to the full path of git.exe for "Script to run"; don't just enter the text "git.exe" 

You can find out what the path is by Tools > Options > Git tab > bottom area called "Git Version"

This workaround does not work for me. Here is the output:

git stash push <cut out filenames>
Completed with errors, see above.

What errors above?

If we use the Custom Action solution above (which does work), then is there any way to name (or rename) the stash?

Seems like this would be a very useful feature!

@L go to Tools > Options
Select the Custom Actions tab.

Can't see the image but yeah it doesn't exists for me. And yes its windows10.

menu

@L this doesn't exist for you? Are you on Windows?

@bablex Thanks - this is a good workaround!

Small addition though:

Tools => Options => Custom Actions => Add

Then just select the files you want to stash (hold ctrl to select multiple).
Right click on any file => Custom Actions => Stash selected files

This doesn't work for me (might be a bug in the latest version?) - there is no submenu of "Custom actions" if I right click. For this I have to go into the  menu

Actions => Custom Actions => Stash selected files

There is actually a simple workaround for this.
Tools => Custom Actions => Add

"Menu caption": Stash selected files
Uncheck all checkboxes EXCEPT "Run command silently"
"Script to run": git.exe
"Parameters": stash push $FILE
(if you want to keep staged changes, add  --keep-index to parameters)

Then just select the files you want to stash (hold ctrl to select multiple).
Right click on any file => Custom Actions => Stash selected files

Now it will only stash the selected files.

It would be really helpful if this feature gets implemented. Thanks guys...