PLT OF BE58 BARON LNDG AT NIGHT IN POOR VISIBILITY WITH RAIN; RUNS OFF THE END OF THE RWY XING THE LIGHTS; ONTO THE OVERRUN.

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I TAXIED A BARON OFF THE END OF RWY 20R AT BNA. I CHOSE TO LAND ON THE R SIDE; AFTER THE TWR OFFERED A SIDESTEP TO RWY 20C (WHICH I COULD NOT SEE). IT WAS AN UNEVENTFUL LNDG. I DIDN'T EVEN USE THE BRAKES; I JUST LET IT ROLL OUT. THE TWR CLRED US TO TAXI TO OUR RAMP. THE FIRST LOGICAL THING THAT CAME TO MIND WAS TO TAKE RWY 31; THEN TXWY T6 TO OUR RAMP. I FELT AS IF WE WERE APCHING RWY 31; BECAUSE I WAS NOT AWARE THAT WE HAD ALREADY PASSED IT. I SUDDENLY SAW UNLIT LIGHTS (THE APCH LIGHTS TO RWY 2L) FACING THE OPPOSITE DIRECTION; APCHING IN FRONT OF US ON THE RWY. IT WAS TOO LATE; DUE TO THE REDUCED VISIBILITY; I TAXIED ACROSS THE LIGHT SYS ONTO THE OVERRUN. WE STOPPED ABOUT 10 FT FROM WHERE WE CROSSED THE LIGHTS. I IMMEDIATELY CONTACTED THE TWR AND TOLD THEM ABOUT OUR SIT. HE CONTACTED ARPT OPS WHO BROUGHT SEVERAL TRUCKS OUT AND TOOK SOME PICTURES. I INSPECTED THE ACFT FOR DAMAGE. THERE WAS NO EVIDENCE OF A PROP STRIKE OR DAMAGE TO THE GEAR. ARPT OPS HELPED ME PUSH THE PLANE BACK ONTO THE RWY; AT WHICH TIME I TAXIED TO OUR RAMP. I NOTED THE OCCURRENCE IN THE ACFT MAINT LOG. A MECH SWUNG THE GEAR AND LOOKED FOR DAMAGE WHICH HE FOUND NONE. I REALIZE THAT IF I HAD THE ARPT DIAGRAM ON MY LAP; THAT THIS MAY NOT HAVE HAPPENED. THERE WAS RAIN ON THE WINDSHIELD AND OUR VIEW WAS EXTREMELY LIMITED. I WAS SO PREOCCUPIED WITH FINDING RWY 31; THAT I MUST HAVE MISTAKEN THE OVERRUN (WHICH HAS NO LIGHTS FACING IN MY DIRECTION) AS THE CLR; BLANK; AND EMPTY VOID THAT ONE SEES WHEN APCHING AN INTERSECTING RWY IN THE DARK. TWR ASKED ME TO CALL HIM ON THE PHONE. I CALLED HIM AND HE ASSURED ME THAT IT WAS NO BIG DEAL AND NO ACTION WOULD BE TAKEN; BUT ARPT OPS INFORMED ME THAT THEY WOULD SUBMIT A RPT TO OUR FSDO. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATES THAT HE THOUGHT THE ACFT WAS AT THE INTXN OF RWY 20R AND RWY 31. THEY ACTUALLY HAD LANDED AND ROLLED TO THE END WITHOUT REALIZING THE MISTAKE. HE HAD NEVER SEEN A RWY WITH NO END LIGHTS AND WAS VERY SURPRISED TO BE THERE. IT WAS RAINING VERY HEAVILY AND THE VISIBILITY WAS QUITE LIMITED. THE ONLY DAMAGE WAS TO 2 TIRES; SOME SMALL NICKS. HE RPTED THE INCIDENT TO THE CHIEF PLT AND WAS GIVEN SOME GND SCHOOL TRNG IN RWY LIGHTING. TO DATE HE HAS HEARD NOTHING FROM THE FAA.