Yes, believe it or not, there is a company that will “lease to own” puppies and dogs ! For a link to the story, click here.
Regardless, if you are leasing a dog, or more traditional things like vehicles, couches, computers or time shares,  lease or rental contracts can be “rejected” in bankruptcy.  That means that you can “get out of” the lease or contract and not owe any past or future fees or costs. Of course, you will have to surrender back to the company the item you are leasing or renting, but often times that is a much better option than remaining liable for the debt. Leases or contracts can be “assumed”, which means you can keep paying for the item and keep it, even though you are filing bankruptcy. You may want to do that, for example,  in the case of a vehicle that is necessary to keep as it may be your only vehicle and you can afford the lease payments.
Back to the dogs !  I would suggest a better option for getting a new puppy is visiting your local animal shelter.  That way, you are giving a needy pup a home and you won’t have to part with him if you do file bankruptcy !
If you are in a lease or rental situation and want to learn the effect filing bankruptcy will have on that lease or rental, be sure and discuss your situation with an experienced bankruptcy attorney … and be sure and mention to them if  you are renting a dog !