A P2P system does not have servers. Instead it has a network of nodes that each as as important as any other node.
Email is perfectly capable of operating in this manner. If everyone ran their own SMTP server, then it would be "fully" P2P. However, most people prefer to use a centralized service. The system is still P2P, though, since it's entirely possible for two people to communicate directly.
In the future most Bitcoin users will transact through MyBitcoin-like services, and running a full network node will require gigabit-speed Internet access. Does that make Bitcoin non-P2P?
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In a server based system like current email if the server goes down then there is no way for the network to function.
If one SMTP server goes down, then the rest of the network functions just fine. Mail will even be delivered after the server comes back up, since other peers will keep trying to deliver mail for several days.