@opps reminded, in another thread, that smoke alarms should be replaced every 10 years. This is true regardless of whether the alarm tests successfully.
After unattaching the smoke alarm from the wiring and turning it over, and partially breaking the key slot when I tried to push a screwdriver into there, I finally got it by using a smaller screwdriver to pry up a long-ish piece of plastic on the bottom, and pulling the drawer out at the same...
Thanks for all the comments here. I got it by turning the smoke alarm over and putting a screwdriver underneath the edge of that panel, as folks are calling it, to pry it up. Hopefully, I'll be able to attach a helpful picture to this post.