Check out this section in the Inertia docs about Persistent layouts: https://inertiajs.com/pages#persistent-layouts
You're going to want to render the "floating" component in there so it persists across multiple page visits. Maybe even portal it in there using portal-vue.
I highly recommend playing with the demo "Ping CRM" app for Inertia: https://github.com/inertiajs/pingcrm
Once you have that setup, add the following as the last element before the closing div in Layout.vue
<div class="fixed bottom-0 right-0 z-50">
This is a floating player
</div>
Now click around to other pages and you'll notice that this element remains there because it's being rendered in the Layout component instead of a specific page.
As for the floating player, that could differ based on what type of player you're using. I'd imagine just rendering it in a smaller size should work?
Hope this helps.