I'd argue any front-wheel drive van with a good quality set of all-terrain tyres would cope with this type of use. Given your budget, the Ford Transit Connect would be my recommendation although petrol models are rare.

That said, if you do need something that's more capable for off-roading, I'd suggest the Peugeot Partner with Grip Control. They are difficult to find on the used market because they sell in low numbers when new. However, they are very good, adding underbody protection and utilising a clever traction control system that allows the front wheels to cope with mud, sand and slippery terrain.