Is there any way to export a set of captured UDP packets as separate files with just the UDP payload?

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Maybe tshark can help a little with that, by using -Tfields and -e to export the data.data field, which will contain the colon separated hex values of the UDP payload, e.g.

tshark -r test.pcapng -Tfields -e data.data

You would probably need to convert the ascii hex stuff back to something binary afterwards, but it could do what you need.

Having Wireshark spit out individual files per frame is not something it can do (although for a datagram type protocol that wouldn't be strange thing, in order to preserve message boundaries).

One way to do this is to export the data as a text file (e.g., in JSON or PDML) and post process from there.