Not fixed while in operation, no (if it's not rigidly mounted it can't apply torque to the hull). Not fixed permanently, yes: they're called "daggerboards."
“Fixed” refers to the fact that the keel isn’t removable. It’s either part of the boat’s hull, or it’s bolted on and can’t be removed without dry-docking the boat.
There are some boat designs with removable keels. For instance, many small sailboats (called a dinghy) have removable keels, so you can remove the keel and beach them easily. The downside to having a removable keel on a sailboat is that it makes the boat prone to capsizing in strong winds; A fixed keel can have a built in ballast to keep the boat from tipping. But on a dinghy, you need to use a live ballast (literally your own body weight leaning out of the boat) to counteract the wind’s effects.