The most common and most likely answer to this crossword clue is the 5 letter word DUCHY
D
U
C
H
Y
More clues leading to the result DUCHY
Buddy explains!

The straight part
We know that "Lancaster, for example" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lancaster, for example = duchy.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does DUCHY mean?
DUCHY noun- A dominion or region ruled by a duke or duchess. (A grand duchy may be a self-governing state. A simple duchy tends to be a part of a larger kingdom or empire.)