If you are in France, or around Europe and don't mind sending your Pi via mail to them, Faimaison and Tetaneutral do propose small computers hosting in their datacenter racks, Pi type included, but also NUCs, respectively for 24€/month (bit expensive but small structure compared to Tetaneutral edit: it's ~15€/month nowadays) and 5 to 10€ / month. That's just an example. Generally you'll get one IPv4 and one /56 or /64 IPv6 prefix.

You might want to look near your location if there is a LUG, non-profit ISP, or non-profit colocation proposing the same kind of services. You may even meet some nice people! But it's definitely doable at least in Europe.

I don’t know of anywhere that will put a Pi you send them in a data centre, but Mythic Beasts do hosting on a dedicated Pi if that helps you at all ?

I found this list from last year. Some of these are more expensive than a small VPS, and I wonder what happens when the Pi needs a reset, or when you want to install a new OS.

If you are located in the US, I have some data center space. Reach out to me and we can talk about it if you're interested.