A two month old Labrador puppy would do fine here in either area. The thing about the smaller area is there are not usually any other dogs for your pup to play with, good area for one on one with your pup. I think the larger open area in the back and the area leading up to that is very good. Most dogs will play according to the size and age of the newer, younger dog. Everyone keeps a watch on the dogs so I feel your dog would be safe and really enjoy the socialization. You can keep a leafd/leash with you but let your pup walk free (I believe a pup in a leash around unleashed dogs is a little bit of a target) and always attach if you are feeling nervous.
There is a play area for smaller dogs. It is not necessarily for younger ones, so a puppy may be playing with an geriatric miniature schnauzer.
Specifically for a two month old Labrador puppy, no. But there are two gated areas. One for bigger dogs and one for smaller ones. Although you can take the smaller ones into either.
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