The most common and most likely answer to this crossword clue is the 8 letter word ENGRAVED
I've seen this crossword clue in The Times
E
N
G
R
A
V
E
D
Buddy explains!

The straight part
We know that "Deeply impressed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case deeply impressed = engraved.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 ENGRAVED mean?
ENGRAVED verb- To carve text or symbols into (something), usually for the purposes of identification or art.
- To carve (something) into a material.
- To put in a grave, to bury.
- Made by engraving
- Decorated with engravings