Yes you can! The Texas A & M AgriLife Extension has directions for freezing radishes. Freezing radishes will alter their texture slightly as they are composed of mostly water. Cutting and blanching them prior to freezing helps to minimize this unwanted effect. Here’s how:
Scrub radishes in cold water; Cut (or grate) into smaller pieces.
Blanch in boiling water for 2 to 3 minutes.
Transfer radish pieces to an ice bath; drain well.