Clue: Quiet time surrounded by deer, dogs and ponies (9)
Answer: SHETLANDS
Explanation: Quiet time surrounded by deer, dogs and ponies (9)
**SH** ("quiet") + **T** (time) *surrounded by* **ELANDS** ("deer") 
*Elands are antelopes, but are also known as Eland Deer, and the word "eland" comes from a Dutch word for "elk" or "moose", which in turn comes from a Lithuanian word for "deer".*
 
The solution refers to Shetland Sheepdogs (aka Shelties) and Shetland ponies.