Is Horseradish deer resistant?

The heat that makes the root worth grating is exactly what keeps deer off the leaves. Any fragment left in the ground regrows, so plant it somewhere permanent.

Do deer eat horseradish?

Rarely. Rutgers rates Armoracia rusticana Rarely Damaged, A, and Iowa State lists it in the top tier of herbs; NC State's entry says nothing about deer at all, so the tier rests on those two. The mustard oil that makes the root worth grating is in the leaf as well, and the leaf is coarse and large. Nothing consulted here records a deer choosing it.

Where it grows

NC State: zones 4a to 9b, 2 to 5 feet, full sun or partial shade, white flowers in spring, and one problem, "Weedy." The warning is worth quoting in full: "If left unharvested the plant will spread by underground roots and become weedy forming colonies," "The smallest piece left in the soil can take root and grow into another plant," and "They can be difficult to eradicate." A permanent bed, or a bottomless tub sunk in the ground.

Protecting horseradish

Nothing. The question is containing it.

If it is browsed

It regrows from the root, which regrows from any fragment, which is the whole difficulty with the plant. A browsed horseradish is not a horseradish anyone will notice.