If you want it to be perennial then just the side shoots. You don't say where you are in the UK but if, like me, you are in a not very warm area then it may be as well to grow it as an annual and cut it all. I have tried over wintering in the cold green house but even that is not very successful. If you don't want to eat it all immediately then, if you like it gently stir fried in butter, then freezing is fairly successful but I prefer to pickle any excess.

I thought this was a seashore plant - is this a different one and if not does it need a salty soil?

It is a seashore plant Steragram. Grows wild on cliffs. I am of course by the sea and it certainly enjoys the salt air and winds. I suppose it doesn't mind a salty environment but I grow it in everyday compost watered with rain water with an occasional spoonful of sea salt in, just the stuff we use in the salt mill from the supermarket. It is delicious with white fish just buttered or pickled in white wine vinegar. Used to be a peasant veg because it could be collected for free and is now available in very trendy restaurants but definitely not for free, lol.