I am wondering about the deadly effect during a military challenge.

Let's say I defend, unsuccessfully, with one non-deadly character against a character with deadly.

The rule manual states that the deadly effect takes place after the challenge resolves. Can I choose my defending character to be killed as the claim for the challenge, leaving no participating characters to be killed by the deadly effect?

yes, you kill your character for claim first and if there is no one left after that then deadly doesn't have anything to work on.

This is the reason that Deadly in a military challenge is usually pretty meaningless. Now, Deadly in an intrigue or power challenge....