Sounds like leaf miner. It's a tiny little worm inside the leaf feasting away. The brown patches show his path of destruction. Cut off the infected leaves and discard. I don't recommend any kind of pesticide. You can eat the leaves like spinach, just make sure they are clean & pest free. Luckily beets don't require much time to mature. Harvest your beets as soon as possible.

I think that you have a problem with Boron deficiency in your soil. You can Google Boron deficiency in beets (beetroot) and you will find, that it matches your problem. It looks like Borax could be the answer. But first see what you think?

Thanks both of you for very different solutions! We have for the moment cut away some of the brown parts. We have some more beets in another plot quite close and they are not affected. Hopefully the unlucky ones will be edible.

That sounds like the best solution. Keep cutting away the affected parts and keep the plots clean as possible to maximize sun & airflow around each plant. That alone would deter these pests.