Animated GIF with (partial) transparency? No.
But I can do you one better anyhow: Animated PNG
So here's how to do it...
Save your PS animation out as a PNG sequence by going to File > Export > Render Video
Choose export folder, then select 'Photoshop Image Sequence'
Make sure to enable the alpha channel as well. You'll want 'Straight - Unmatted'.
So now that you have your image sequence – here are some of your options for assembling them into an APNG:
As for support – Chromium (and therefore other Chromium based browsers such as Vivaldi and Google Chrome), Firefox, Opera and Safari browsers all now support APNGs:
The only major browsers lagging behind are (as usual) Internet Explorer and Edge... As we've seen – the world does not wait for Microsoft.
Now we'll see this format take off for sure and here's why:
Here is an example I made with (full and partial) transparency – first saved out as an animated GIF and lastly as an animated PNG:
Just a wee bit of a quality difference eh? And the animated GIF is 66kb while the animated PNG is only 45kb...
No contest... game over.