Strip of woodland ownership

There is a narrow strip of woodland that has a footpath running through it near to where I live. I have maintained this area by clearing the footpath and digging the ditches out either side, as well as clearing any wind felled trees. All locals I have spoken to say the woodland is not owned by anyone. The property does not come up when using the map on the government website to search for land or properties. I have attached a screenshot of an OS map to show the strip of woodland. How do I go about finding out if anyone does own it, and if not is there a way to purchase/claim it (adverse possession?, although this may be difficult due to the footpath?) I have been maintaining it/using the woodland for around 6 years now. Closest postcode is SN15 5PG.

Joe - all land is owned but not all land is registered. As such I would recommend starting with checking if the land is registered as that will impact on how any 'next steps'. If you've tried online using the search by map option, and had no luck, it may well be unregistered. But you should apply by post for a search of the index map for certainty https://www.gov.uk/get-information-about-property-and-land/search-the-index-map