I don't think you're likely to find this in a padlock, but I've understood a disc detainer lock can be made so that a key of one bitting is needed to open the lock, and a different bitting to lock it.
In a disc-detainer lock, the key operates the discs when it turns (instead of when it's entered in the keyhole, as in pin-tumbler lock), so opposite sides of the key can have different bittings. Combine that with a cylinder lock that moves the deadbolt in one direction when turned clockwise, and in the other when turned counter-clockwise, and you get the possibility of open-only and lock-only keys.
Now, I've never seen such an asymmetric key in real life, but apparently they're called shopkeeper keys, and are (or have been) made at least in the Abloy Disklock and Protec systems. Disklock is an older system, and toool.nl has a document with all you'll likely want to know about it, including a mention of the shopkeeper function. A quick web search found Protec cutaway locks with that feature being sold, too.
The idea (and what the name refers to) is that you can leave a close-only key to an employee on evening shift, and the key can't be used to get in after they left. (Well, unless they forget to lock up.)
As for the public-key cryptosystem analogy, I'm not sure if a shopkeeper key like that makes sense. A public key is supposed to be known by everyone, so they can send you messages, but handing close-only keys to random guys on the street would quickly get some joker locking your doors. (With your employees being unable to open them...)
The analogy of a safe or strongbox that can be locked without a key and sent back to its owner seems to make more sense, if anything will. (Or perhaps a mailbox-like box with a slot you can drop messages in.)
