The most common and most likely answer to this crossword clue is the 9 letter word PACHYDERM
I've seen this crossword clue in Independent
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More clues leading to the result PACHYDERM
- Thick-skinned type, my padre, crazy about church
- Insensitive sort? My dear chap, ignoring an American is dreadful
- Elephant maybe taking short step before my herd goes berserk
- Thick-skinned beast, Mr Hyde on rampage after cover-up
- In cramped condition heartless Henry is an insensitive type
- Insensitive person vacantly read my cheap novel
- Jekyll’s alter ego gripped by terrible cramp sees a hippo or elephant
- Marc hyped arrangement with an insensitive person
- Elephant capsule, in retrospect, transformed Mr. Hyde
- Thick-skinned individual marched awkwardly around yard after start of parade
Buddy explains!

The straight part
We know that "Insensitive sort?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case insensitive sort? = pachyderm.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 PACHYDERM mean?
PACHYDERM noun- A member of the obsolete taxonomic order Pachydermata, grouping of thick-skinned, hoofed animals such as the rhinoceros, hippopotamus, elephant, pig and horse.
- An elephant
- A person with thick skin; someone who is not affected by or does not care what others say about him or her.
- Someone who is insensitive.