A second later, it clicked: we are in Australia! and there is A SPIDER by my pillow!
I peered cautiously, as much as I could at a safe distance, at the huge creature, and after much scared and panicky but slightly curious standing around staring at it, we deduced it was a Wolf Spider. They can give a nasty bite apparently, but thankfully not fatal.
Still, it was not what we wanted by the bed.