Okay, your problem is consistent with a failure in the brake light circuit, i.e. the switch.
Now I don't know which type your car uses, so I'll offer a generic troubleshooting for both the mechanical(on the pedal stop) and hydraulic.
First you need to id which type you have. Maybe a parts book or a friend, or just looking under the dash at your brake pedal.
If it's the mechanical type, they are usually mounted at the brake pedal stop, a bracket that the pedal rests up close to when the pedal is not depressed. You should see a small cylinder with a couple of wires on it. One is battery voltage and the other goes out to the brake lights.
If so, push in and out on the pedal a few times and see if the switch is being shut off completely, usually a small button inside threaded portion of the switch. A lot of times the switch can be adjusted by loosening the locking nut and turning the switch in(to shut off sooner) or out for desired pedal shut off(make sure you unhook the wires while turning the switch, then reinstall them).
If not this type, then out in the hydraulic circuits by the master cylinder(follow the piping from the master cylinder) you will find a little stubby switch with two electrical connectors on it. This is the hydraulic brake light switch. No adjustment, just replace.
Way to determine, is get a voltmeter or test light. Test the wires going to the switch(either type). One wire should have battery voltage to it whenever the fuse is energized(some cars you have to turn the key on, others it's on all the time). Then test the other wire, should be no light, or voltage, UNTIL you press on the brake pedal, then it should light up. Way to temporarily stop battery draw,if the light is staying on, is to unhook the output terminal of the switch. Sometimes easy to get too.
HTH