This is basically a note to myself: I want a bit crazy searching for where the "delivery slips" were stored in the PrestaShop database. It turns out that there is no "delivery slip" table, rather the delivery slip ID is actually part of the ps_order_invoice table (field delivery_number). The value is filled out whenever an order is set to "Shipped" or "Processing in progress".
Which basically means that PrestaShop is not set up for making multiple shipments on the same order. Not very handy, especially for B2B orders.