Export Photoshop Vector with Clipping Mask to SVG

Hello!

Does anyone know if it's possible to export a Photoshop vector with clipping masks to a SVG (or other universal vector format) correctly? My art files are made up of vector shapes and clipping masks. The vector shapes export correctly but the masks seem to get lost. I will attach some screenshots to show what I mean. For the life of me I can't figure out a way to covert these Photoshop vectors into a universal format. I've tried EPS, PDF, and opening the PSD in Illustrator, too, but all of those methods embed the art as images instead of exporting the vector shapes.

Here is an image of my clipping masks in Photoshop:

Here is the resulting SVG file - notice the clipping masks are lost:

Here is the how the files opens in Illsutrator as a PSD, EPS, or PDF - notice how it embeds images instead of creating proper vector shapes:

Thank you in advance for your help!