Do soldiers lose Will while they recover from injuries?

no if they were critically wounded they lose a huge amount of will. I think

If they are critically wounded (IE: they would have otherwise died, have to be stabilized or win within 3 turns), they will permanently lose Will. If they suffer injuries but don't "die", they will not lose Will (at least not permanently).

A soldier who gets a regular wound will not lose will permanently though they will suffer a loss of will during the battle in question where they got it. Once the battle is over the will returns.

If a person gets critically wounded then yes, the loss of will is permanent but be aware that there is a difference between a critical hit and a critical wound if you didnt know this.
A critical hit, as you probably know is just getting hit by the enemy and you get additional damage while a critical wound will leave your trooper on the ground hovering between life and dead while needing either the mission to end before he/she dies or medical treatment from a medpack.