Yes, the larger ones do. They prefer to confine problem-guests to their own quarters. But for the worse cases, the larger ships do have a brig (or they can improvise one). It’s often a steel-walled room or rooms or barred space near the security office.
Here is one example.
I imagine it “feels like” you just blew your fare and being concerned about what the next port of call’s justice system is liable to do with you.