B737-300 CREW ENCOUNTERED NIL BRAING IN THE LAST 1200 FT OR SO OF RWY 23L AT ZZZ. THE ARPT HAD PAINTED THE LAST PART OF THE RWY.

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PERFORMANCE DATA WITH 30 DEG FLAPS; WET-GOOD; SHOWED NO BRACETS WITH MINIMUM BRAING. RWY -- GROOVED; WET BUT NO STANDING WATER; TOLD RWY MARINGS WERE VERY RECENTLY PAINTED; NO REVERTED RUBBER NOTICED (POSSIBILITY REMOVED IN CONJUNCTION WITH REPAINTING). WE WERE BEING VECTORED FOR ILS RWY 23L. ABOUT 20 NM FROM FIELD; WE WERE ASED TO SLOW TO 160 TS TO FOLLOW AN RJ THAT WAS BEING VECTORED FOR A VISUAL APCH TO RWY 23L. ABOUT 17 NM; WE WERE REQUESTED TO SLOW TO SLOWEST POSSIBLE SPD. WE SLOWED TO OUR APCH SPD (140 TS). AT ABOUT 3000 FT MSL AND 10 NM OUT; WE ENTERED VMC AND WERE ABLE TO MAINTAIN VISUAL WITH THE AIRFIELD. AT ABOUT 8 NM; WE SIGHTED THE RJ THAT WAS ON A L VISUAL DOGLEG TO FINAL. WE WERE CLRED THE VISUAL APCH TO RWY 23L BEHIND THE RJ. BECAUSE SPACING WAS TIGHT (ABOUT 2.5 NM); WE RECEIVED CLRNC FOR AN S-TURN TO THE L (E) OF FINAL. S-TURN MANEUVER WAS COMPLETE BY ABOUT 1200 FT TO 1300 FT AGL. APCH SPD AND GLIDE PATH WERE NORMAL; AND STABILIZED THROUGHOUT S-TURN. ABOUT 3 NM SPACING WAS OBTAINED AND RJ CLRED THE RWY. ALL STABILIZED APCH CRITERIA WERE MET WELL BEFORE 1000 FT AGL AND MAINTAINED THROUGHOUT THE APCH. TOUCHDOWN WAS NORMAL ABOUT 1500 FT FROM THRESHOLD. NOSEWHEEL WAS LOWERED NORMALLY. NORMAL REVERSE THRUST (NOT MAX). NORMAL BRAING (NOT MAX) INITIATED AT ABOUT 80 TS AND REVERSE THRUST STARTED TO BE REDUCED. ACFT DECELERATION APPEARED NORMAL. ON ROLLOUT; I CONSIDERED TURING OFF AT A2 INTXN (SECOND-TO-LAST TURNOFF). IT WAS MY JUDGEMENT THAT IT WAS POSSIBLE; BUT WOULD HAVE REQUIRED HIGHER THAN NORMAL BRAING; SO I DECIDED TO PROCEED TO A1 (THE END TURNOFF). BRAING ACTION AND ACFT DECELERATION WERE STILL NORMAL. CAME OUT OF REVERSE THRUST AND CONTINUED TO INCREASE BRAE PRESSURE FOR NORMAL TURNOFF AT A1. AT ABOUT 1500 FT FROM THE RWY END; ANTI-SID STARTED CYCLING AND BRAING ACTION APPEARED TO DECREASE. I MAINTAINED BRAE PRESSURE; BUT IT SOON BECAME OBVIOUS THAT DECELERATION RATE WAS NOT AS GREAT AS I ANTICIPATED DUE TO THE ANTI-SID CYCLING. I INCREASED TO MAX BRAE PRESSURE. I DECIDED TO TRY TO STOP STRAIGHT AHEAD INSTEAD OF TRYING TO MAE A HIGH SPD EXIT ON A1 (TURNOFF IS A 90 DEG; NOT A HIGH SPD EXIT) IN ORDER TO MAX BRAING. SPD WAS SUCH THAT ACFT WAS NOT GOING TO STOP COMPLETELY ON THE RWY. THE RWY HAD RAISED LIGHTING AT THE END OF THE RWY (THRESHOLD LIGHTS FOR RWY 5R). HOWEVER; THERE WAS SUFFICIENT SPACING BTWN THE LIGHTS. I MANEUVERED THE ACFT SO THE NOSEWHEEL DID NOT CONTACT ANY LIGHTS. ACFT WAS STOPPED WITH THE NOSEWHEEL ABOUT 25 FT INTO THE OVERRUN. MAIN WHEELS NEVER LEFT THE RWY SURFACE. AFTER ACFT WAS STOPPED; THE FO INFORMED THE TWR OF OUR SIT. THEY ASED WHETHER WE WANTED THE EMER EQUIP. WE STATED NO AND WE DID NOT DECLARE AN EMER. HOWEVER; WE TOLD TWR WE WOULD NEED A TUG FOR A PUSHBAC AND WE WOULD COORDINATE WITH COMPANY OPS. SUPPLEMENTAL INFO FROM ACN 627414: THE ENTIRE PROCESS SEEMED COMPLETELY NORMAL UNTIL WE APCHED THE DEP END OF RWY 23L. WE WERE NO LONGER INDICATING ANY AIRSPD (I WOULD ESTIMATE APPROX 30 TS GND SPD) AS WE REACHED THE AIMING POINT MARER FOR RWY 5R. THE BRAING ACTION WAS DEGRADED TO NIL. THE ANTI-SID ACTIVATED AND PULSED REPEATEDLY UNTIL THE ACFT CAME TO A STOP APPROX 20-25 FT INTO THE RWY OVER-RUN.