IN OVERRUN AREA ACCOUNT NO BRAKING AT TURN OFF EXIT.

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LNDG RWY 22 WILMINGTON; OH. BRAKING ACTION RPTS WERE RPTED FAIR TO POOR UNTIL THE LAST 2000 FT OF THE RWY WHICH WAS POOR AND VERY POOR BY OTHER PLTS AT TXWY A5 WHICH IS THE LAST TURN OFF OF 22 AT THE APCH END OF RWY 4. ON THE MORNING OF OPS; A-5 WAS THE ONLY TXWY OPEN TO EXIT RWY 22. THE RWY WAS COVERED WITH DRIFTING SNOW AND PATCHY ICE. UPON TOUCHDOWN WE HAD FAIR BRAKING ACTION WHICH GRADUALLY DECREASED TO POOR THE LAST 2000 FT OF THE RWY. RWY 22 IS 107000 FT LONG; 150 FT WIDE. THE LAST 1000 FT WE SLOWED TO A CONSIDERABLY SLOW TAXI SPD AND TO ALMOST A STOP AT A-5; JUST ENOUGH SPD TO TURN THE NOSEWHEEL. ABOUT 20 DEGS INTO THE TURN; I FELT THE NOSE WHEEL STEERING GIVE ONE WAY AND LOST ALL BRAKING. NO PWR WAS APPLIED INTO TURN; THE ACFT WAS AT IDLE PWR. THE NOSE OF THE ACFT ENTERED THE OVERRUN OF RWY 22 BY ABOUT 10 FT. NO LIGHTS WERE HIT AND THE ACFT NEVER LEFT PAVEMENT. WHEN THE ACFT CAME TO A STOP; WE WERE STILL IN A POS TO EXIT THE RWY AT A5. BUT; BY MOVING THE NOSEWHEEL STEERING; I COULD TELL WE WERE STILL SITTING ON ICE. WE IMMEDIATELY RPTED BRAKING ACTION AS NIL AT TXWY A5 AND RWY 22 AND REQUESTED A TUG TO TOW US INTO A-5. THE TUG PULLED US INTO A-5; AND WE TAXIED TO THE GATE. NO DAMAGE WAS DONE TO THE ACFT NOR ANY RWY OR TXWY LIGHTS OR SIGNS. IN REVIEW OF ALL THE CIRCUMSTANCES; I FEEL I DID EVERYTHING POSSIBLE GIVEN THE RWY CONDITIONS. I SLOWED THE ACFT TO CRAWL WHILE EXECUTING THE TURN. ONCE IT HIT THE ICE; THE ACFT SLID. REVERSE AND THE SLOW SPD STILL DIDN'T PREVENT THE ACFT FROM GOING 10 FT INTO THE OVERRUN AREA. AGAIN A5 WAS THE ONLY EXIT OPEN ON RWY 22.