But in the case of gallium arsenide, the arsenic is a component of the pure (undoped) crystal.
Oh. I did not know that. So does that mean that after doping, both the p and n regions still have arsenic?
Yes, exactly. GaAs is a III-V compound, it's elements come from the group III and group V columns of the periodic table. The usual silicon dopants Boron, Phosphorous, Arsenic, Antimony just drop in in place of the Galium or Arsenic.
To dope GaAs "P" you use group II - zinc, or for "N" group VI Sulfur, Selenium, or Tellurium.
It was lots of fun making LEDS with GaAs, and GaAsP!