The only climbable blocks in Minecraft are ladders, vines, nether vines, and scaffolding, not including liquids because it is also impossible using those. Therefore, the death message “player fell while climbing” is impossible to get.
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violine1101
It actually is possible XD
As far as I'm aware, you can get if if you modify a block that is usually not climbable to be climbable using a data pack.