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

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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.

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.