Looking for animals dataset for Deep Learning classification
Unfortunately, a single dataset with all animals does not seem to exist (perhaps you can make one :D ), but there are plenty of datasets with a subset of animal species. Here are a few I can think of:


There are many datasets on Kaggle. Searching for "species", "animal", or some other smarter keyword should give some options. I found the 10 Monkey Species, STL-10, bird species classification, and Animals-10 datasets fairly quickly, for example.
Most large-scale datasets like OpenImages, CIFAR, ImageNet, the Visual Genome, and COCO have animals as some of the categories (among non-animal ones).
The iNaturalist dataset is a large scale species classification dataset (see the 2018 and 2019 competitions as well). 
The Nature Conservancy Fisheries Monitoring dataset focuses on fish identification.
The Caltech-UCSD Birds-200-2011 is a standard dataset of birds.
The animals with attributes 2 dataset focuses on zero-shot learning (also here).
The Snapshot Serengeti dataset covers 40 mammals from the African Savannah.
The Oxford-IIIT Pet Dataset is good for fine-grained cat and dog classification.
A survey on image-based insect classification by Martineau et al has references to a number of image datasets for insect identification (though I am not sure how many are public).
The group at the Labeled Information Library of Alexandria: Biology and Conservation (LILA BC) track a large set of animal conservation and ecology datasets, including many involving image-based species recognition here.