Mutable internal value in discriminated union

Question

The task is to create a binary tree with its nodes having a member is_locked that can be true or false and a method lock() that shall set is_locked to true if it's not already.

I tried this

type BinaryTreeNode =
    | Node of BinaryTreeNode * BinaryTreeNode
    | End
    with
        let mutable internalIsLocked = false
        member this.is_locked
            with get() = internalIsLocked 
        member this.lock() = internalIsLocked <- true

but compiler says no due to This declaration element is not permitted in an augmentation at the let binding.

val mutable internalIsLocked : bool instead does not work for the same reason.

How can this problem be solved then? Is it not solvable with discriminated unions?