Rangers have found the bodies of several animals in recent months that have been run over by vehicles in Yala park, home to the world's highest concentration of leopards and large numbers of elephants, bears and deer.
"When a leopard or other interesting sighting is made by one vehicle, the news is rapidly transmitted by means of mobile phones, attracting large numbers of vehicles to the site," the government's department of wildlife conservation said in a statement announcing the ban.
Yala, which covers an area of around 985 square kilometres, runs along Sri Lanka's sparsely populated southeastern coast.
It attracts more than 100,000 foreign tourists each year and is a key source of revenue to the government.
Sri Lanka was the first country in South Asia to introduce cell phones in 1989 and there are now some 22.12 million registered mobile phones in a country with a population of just 21 million.