Oh! Sorry, I didn't read your post correctly... I thought it was the E-brake indicator, only...
If it's the round brake symbol, that's the ABS system light... and mine would stay on a lot also... it is due to "sensor" failure... (not system failure...)
The ECM (on board computer) gets a signal from the wheel sensor that is not to its liking and powers up the indicator to say there is a problem (somewhere...) in the ABS system... That somewhere can be anything from the firewall on down the system to the metal gears that tells the sensor reads to tell the ECM that the wheels are turning...
Unfortunately, putting the diagnostic reader that you can borrow from Auto Zone or O’Reilly Auto Parts, on the system, will only tell you the same things as the indicator light... "there is a problem with the ABS system..." (Which you know is not true, because the indicator will go off...)
So, more than likely, you are dealing with ONE of the wheel sensors or wheel speed speed indicators, (again, on one of the wheels...) Either the sensor is going bad... (they can go bad for a number of reasons...) Or, they're not bad at all, but... and this is the most common fault... it has picked up iron particles from the road and they are sticking to the sensor (which is magnetic). That fools the sensor into thinking, that wheel(s) is not turning.
But, the ECM knows there is more fuel going into the injectors, the throttle positioning sensor is open, the transmission speed sensor is sending the signal that the transmission is moving... And, that one sensor is not picking up a pulse from that one wheel... This makes the ECM think that one wheel is not turning...
So, here's what it looks like and how it works...

(Ford does not use the Optical type on the F53...)
Now, the fun part... Which wheel is it...? (My money is on the passenger's front...) That's been my experience... But, you have three more to choose from... BTW: That "Air Gap" is usually less than an 1/8th inch and crud can get in there pretty easily..
Oh, and replacing the sensor is usually the cure, even if it is only dirty, because, well... the mechanic is already in there, for $125 - $200 for the repair... And, he figures, for $26 for the part... why not put a new one in...?!?
Cheers!
Bob