If he could only get back to cover, he might find a corner to protect his rear and have some fighting chance for life.

But wherever he turned that huge cat faced him, doubly armed, and equipped as a fox can never be for the snow.
No one could watch that plucky fight without feeling his sympathies go out to the beautiful silver fox.

Rolf, at least, was for helping him to escape, when the final onset came.

In another dash for the woods the fox plunged out of sight in a drift made soft by sedge sticking through, and before he could recover, the lynx's jaws closed on the back of his neck and the relentless claws had pierced his vitals.
The justification of killing is self-preservation, and in this case the proof would have been the lynx making a meal of the fox.

Did he do so?
Not at all.