Wikipedia says yes: https://en.wikipedia.org/wiki/Basalt_fiber

> The basalt fibers typically have a filament diameter of between 10 and 20 μm which is far enough above the respiratory limit of 5 μm to make basalt fiber a suitable replacement for asbestos.

The source mentioned is a basalt fiber brand website, so not sure if that's enough for confidence.

but still it breaks and gets airborne...