Brockswood Animal Sanctuary on Catholic Lane will shut from today until Sunday, October 22.
Bosses said the reopening date could vary, so urged visitors to keep an eye on their social media channels.
No animals or visitors were injured by the fallen tree on Friday afternoon due to being inside their paddocks in the heavy rain.
A spokesman said: “Unfortunately, we cannot open for your safety. One of our very large willow trees fell into one of the paddocks this afternoon, blocking off our main pathway.
“As you can see from the photos, the damage is severe, and we have to ensure your safety before reopening.”
“We want you to know that all the animals are safe. They were in their shelters due to the rain (the one time we are very thankful for the British weather), and our animal care team have been checking on them to ensure they are all okay and not too frightened by this scary ordeal.
“However, this is a big and unfortunately very expensive job, and we need at least a week to sort things out. We must fix the fences and pathways apart from dealing with the fallen tree. We appreciate your patience and understanding during this time, and we are happy to answer any questions you may have.
“Our dog run will still be open as usual throughout the weekend and all of next week. It’s at the other end of the site, so it won’t be affected. If you have any queries, please don’t hesitate to contact us and thank you for your patience and understanding at this time.”