What sport using a ball has the largest playing field?

Polo

Surely golf has got to have a bigger playing area than Polo. They walk for miles in search of their balls.

The outdoor polo field is 300 yards long and 160 yards wide, the largest field in organized sport.
The average 18-hole golf facility takes up 150-200 acres, teeing areas using 2%, putting greens 2%, fairways 23%, rough/woods/water 70%, buildings and grounds 3%.

An acre is equal to 43,560 square feet and a polo field is 432,000 square feet, making the average golf course larger.

Polo has the largest playing field of any organized sport - 300 yards by 160 yards.

http://www.sportpolo.com/sport/polo_field.htm

Hurling is a ball game that is played twice a year in the Cornish Town of St Columb Major. This ancient game is played anywhere within the Parish boundary which is approx 25 square miles.

https://en.wikipedia.org/wiki/Hurling_the_Silver_Ball

Golf is played on a course not a field. Therefor Polo is the answer.

How about Ashbourne Rules football, as in the Royal Shrovetide Football Match? This takes two days and the goals are three miles apart. This is one of the few sports that actually prohibits murder in its rules. (What the penalty is for committing it - apart from the one imposed later in court - I don't know.)
https://en.wikipedia.org/wiki/Royal_Shrovetide_Football