Are you sure it is mice at this stage, my guess it is the 'early birds' i.e. pigeons, jays and a few of the smaller birds.

The new growth is quite succulent to the birds particularly as there is not much about to eat, pea tops make a nice snack.

Netting should be the answer, at least thats what I do

To overcome the 'mouse problem' I sow seed in cell trays, three or four seeds per cell, and germinate in the greenhouse or cold frame.

I then plant out the plugs of seedlings when they are 25mm (1") high,

This method also cancels the possibilty of gaps in rows caused by non geremination.

Hope that helps