UK Oil and Gas made the announcement after it was refused a request for more time to respond to water contamination concerns upheld by the Environment Agency last month.
Campaigners had been due to hold a demonstration outside the planning committee meeting on May 11 for the application, which has attracted more than 2,000 objections.
In a statement to shareholders on Tuesday, UKOG excecutive chairman Stephen Sanderson said: “Given the potential sensitivity of the Markwells Wood site to the adjacent chalk groundwater system, it is in the public interest that this subject be investigated as thoroughly as possible prior to any further site activity.
“Whilst the refusal of SDNPA to accommodate this important step is somewhat unsatisfactory, we look forward to concluding this matter with the EA and resubmitting the application as soon as practicable.
“Our target of drilling Markwells Wood by Q2 2019 is unlikely to be affected.”
The withdrawal has been welcomed by environmentalists.
Brenda Pollack, Friends of the Earth south east campaigner said: “This is fantastic news for local people and this beautiful area. Hopefully UKOG have realised that there is no place for drilling in or under the South Downs National Park.
“The risks to drinking water, increase in trucks on rural roads and potential air and noise pollution were all of concern to Friends of the Earth and many others. It would have had impacts far beyond the village.
“We hope this sends a clear message to other companies that industrialisation of our most precious areas of countryside is not wanted.”
Green MEP for the South East Keith Taylor said: “The triple threat of water supply contamination, wildlife and biodiversity loss, and local community and woodland tourism harm has been averted, for now.”
“It is one small, but important, victory in a region-wide fight against oil and gas drilling plans that will see the beautiful South East countryside pock-marked with wells.”
Markwells Wood Watch, which has protested against the drilling for the best part of a year, has said it will continue to fight for the restoration of ancient woodland and water source protection in the area.
Campaigner Emily Mott said: “Our aquifer is vulnerable and nothing can change that.
“They can do all the tests they like and the conclusions will be the same.”
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