I thought black bodies emitted light the whole spectrum, albeit with differences in the distribution depending on their temperature?

Yep, but the exponential decay at the short-wavelength end means you're going to hit a single-digit number of photons/m^2/s fairly quickly.

> 10–10^3 photons cm^–2 s^–1

So probably invisible but under the very darkest of conditions?

Visible refers to being in the visible spectrum.