How to prevent mice from building nests in the headliner

[some mech 3]

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I no sooner put in a new headliner in my 1931 A coupe than mice got above it and urinated in several places. As a result the headliner was ruined and some staining occurred on the side panels. I have a new headliner but am reluctant to install it for fear that the mice will again ruin it. Is there any full-proof way to prevent the mice access to the area above the headliner? I set about 3-5 traps but they have to be checked often, and traps are not fullproof. Are there any products or coverings that would prevent the urine from seeping through? I'm desperate
Thank you.
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