Solution #1: Using CONNECT BY clause (for range generation), REGEXP_COUNT, and RANK() functions:

WITH w AS (  
SELECT 'arkansas' AS word  
FROM dual  
), r AS (  
SELECT ROWNUM rn  
FROM w  
CONNECT BY LEVEL<=LENGTH(word)  
), x AS (  
SELECT SUBSTR(w.word, r1.rn, r2.rn - r1.rn + 1) substr,  
       RANK() OVER(ORDER BY r2.rn - r1.rn DESC, r1.rn) rk  
FROM r r1, r r2, w  
WHERE r1.rn<=r2.rn  
 AND REGEXP_COUNT(SUBSTR(w.word, r1.rn, r2.rn - r1.rn + 1), '(.).*\1') = 0  
)  
SELECT substr  
FROM x  
WHERE rk=1

Solution #2: Using CONNECT BY clause (for range generation), REGEXP_LIKE, and MAX() KEEP functions:

WITH w AS (  
SELECT 'arkansas' AS word  
FROM dual  
), r AS (  
SELECT ROWNUM rn  
FROM w  
CONNECT BY LEVEL<=LENGTH(word)  
)  
SELECT MAX(SUBSTR(w.word, r1.rn, r2.rn - r1.rn + 1))  
 KEEP(DENSE_RANK FIRST ORDER BY r2.rn - r1.rn DESC, r1.rn) substr  
FROM r r1, r r2, w  
WHERE r1.rn<=r2.rn  
 AND NOT REGEXP_LIKE(SUBSTR(w.word, r1.rn, r2.rn - r1.rn + 1), '(.).*\1')

Solution #3: Using CONNECT BY clause (twice), LATERAL view, REGEXP_COUNT, and RANK() functions:

WITH w AS (  
SELECT 'arkansas' AS word  
FROM dual  
), s AS (  
SELECT SUBSTR(word, LEVEL) word, LEVEL rn  
FROM w  
CONNECT BY LEVEL<=LENGTH(word)  
)  
SELECT MAX(x.substr)  
       KEEP(DENSE_RANK FIRST ORDER BY LENGTH(x.substr) DESC, s.rn) substr  
FROM s, LATERAL(SELECT SUBSTR(s.word, 1, LEVEL) substr  
                FROM dual  
                CONNECT BY LEVEL<=LENGTH(s.word)) x 
WHERE REGEXP_COUNT(x.substr, '(.).*\1') = 0

Solution #4: Using XMLTable function (for range generation), Correlated subquery with COUNT(DISTINCT), and MAX() KEEP function:

WITH w AS (  
SELECT 'arkansas' AS word  
FROM dual  
), r AS (  
SELECT ROWNUM rn, word 
FROM w, XMLTABLE('for $i in 1 to $N cast as xs:integer return $i'  
 PASSING LENGTH(w.word) AS N) x 
)  
SELECT MAX(SUBSTR(r1.word, r1.rn, r2.rn - r1.rn + 1)) 
 KEEP(DENSE_RANK FIRST ORDER BY r2.rn - r1.rn DESC, r1.rn) substr  
FROM r r1, r r2 
WHERE r1.rn<=r2.rn  
 AND r2.rn - r1.rn + 1 =  
 (SELECT COUNT(DISTINCT SUBSTR(SUBSTR(r1.word, r1.rn, r2.rn - r1.rn + 1),  
                               LEVEL, 1))  
 FROM dual  
 CONNECT BY LEVEL<=r2.rn - r1.rn + 1  
 )

Solution #5: Using CONNECT BY, Recursive CTE, INSTR, SUBSTR, and MAX() KEEP functions:

WITH w AS ( 
 SELECT 'arkansas' word 
 FROM dual 
), s(sub, word, lvl, rn) AS ( 
SELECT SUBSTR(word, LEVEL, 1), SUBSTR(word, LEVEL) word, 1, ROWNUM 
FROM w 
CONNECT BY SUBSTR(word, LEVEL) IS NOT NULL 
UNION ALL 
SELECT SUBSTR(word, 1, lvl+1), word, lvl+1, ROWNUM 
FROM s 
WHERE LENGTH(SUBSTR(word, 1, lvl+1))=lvl+1 
 AND INSTR(sub, SUBSTR(word, lvl+1, 1))=0 
) 
SELECT MAX(sub) KEEP (DENSE_RANK FIRST ORDER BY lvl DESC, rn) substr 
FROM s