I'm not aware of anyone having made a core for those.
The small flash absolutely sucks to try to write arduino code with (great lengths were gone to to make the tiny13 usable with only 1k flash) , and programming them is harder because they use something other than ICSP for programming.
They aren't exactly chips that are well suited for use with Arduino.
If is possible- there is a good guide here and the core files are available here. Since there is so little Flash memory it doesn't include a lot of the Arduino functions and you have to use an external programmer since there is no onboard USB (and no room for a bootloader obviously.
This project was started out as simple as all I wanted was to code an LED to blink when a button is pressed and was under the impression that the ATTiny 10 was like the 85 (just upload the code and you're done) but it seems this is going to be a lot more work than I originally thought.
It's always those small projects that end up being the largest lol
You could try contacting the author and ask about Attiny10 support.
As the author of ATTinyCore - No, I will not add support for the ATTiny10. I do not have interest in supporting chips with such awkwardly tiny flash (I stop at 2k, and only support 2k where there's a 4k or larger version of the chip). If I'm going to support a new class of chips with a different programming protocol, it's going to be the new megaAVR ones.
Ask Hans, he does MicroCore, for the t13, it is much closer to what you'd need in order to have a vaguely usable core for it - he has done a lot of optimization to reduce flash consumption.
Have you considered an attiny85 in the MLF/QFN package? Is that small enough, or still too big?
I have no plans adding Attiny10 support to MicroCore. The problem with the ATtiny4/5/9/10 is that they have a rediced instruction set (54 vs 120 on t13). This means the compiler is not able to create as efficient and small code for the t10 as it would for the t13.
The ATtiny13 is kinda capable when used with MicroCore. I'm using a t13 in a class AB amplifier where it's monitoring the temperature of a heatsink with an analog temeprature sensor (LM35DM) and controlling a 12V PC fan with a few speed levels and hysteresis. I'm still only using about half the flash memory available.