Typically not. There will a shift in the pecking order but not as much as an introduced bird will generate. Your egg production will not suffer.

Basically, one of my chickens (a silkie) has died due to old age and I was wondering if her death will impact the egg production of others?

It is clear that when you introduce a new chicken that there is upset within the group so I was wondering if the same happens when a chicken is taken out or in this case dies. (I now have 11 chickens)

Thank you :)