All mammals use a computer: their own brains. However, in terms of analog and digital computers, humans are the only mammals that choose to use computers. Primates and other mammals can certainly be trained to use computers to perform simple tasks (mostly through game-play), however it's doubtful if they gain any real benefit. They certainly don't use them through choice.
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Mammals that use computers would be the following:
Human Beings
Monkeys (In some cases monkeys use simple forms of computers to do different, specific tasks, that are simple to the Human Perspective, but actually quite amazing considering it is a monkey performing the task).