Yeah they will pack absolutely everything they can get their hands on.
We had empty jam jars that had been in a pile for recycling carefully wrapped in bubble wrap, a completely full bin packed neatly in a box, and a half-used packet of bread packed away that we didn’t find for a couple of weeks after the move!
They go really fast and are in robot mode.
Anything that you don’t want packed, I’d move into your bathroom, shut the door, and put a big sign on saying “do not pack!” Maybe with a no-entry road sign as well as they don’t all speak English well.
Key thing while moving with a baby (we moved with a 3 day old, timing issues) was to set up absolutely all the baby stuff as soon as you get to the new place. Set up in the baby’s room with the crib, bottles etc, a chair and drinks for you, and then ignore everything else.
Are you the mum? If so then Really really don’t try to help pack or lift anything, your body is still healing and you can do a lot of damage by lifting heavy things too soon.
Congratulations on your baby and good luck with the move!