  We prove several new versions of Hilbert's basis theorem for non-associative
Ore extensions, non-associative skew Laurent polynomial rings, non-associative
skew power series rings, and non-associative skew Laurent series rings. For
non-associative skew Laurent polynomial rings, we show that both a left and a
right version of Hilbert's basis theorem hold. For non-associative Ore
extensions, we show that a right version holds, but give a counterexample to a
left version; a difference that does not appear in the associative setting.
