Ok, urgent help needed! We have a three-board oak paddock fence, lined with black wire to keep the dogs in.

We are painting the fence today. The idea was to paint the boards and wire along the boards white, and leave the wire in between the boards black so you can't see it well. I've seen fences like this and they look great!

But the paint keeps dripping down the wire, into the spaces between the boards.

Help! How do people do this? There must be some trick I don't know...

Any advice is so very appreciated.