Squares on local rings of characteristic 2

Question

Let $R$ be a local commutative ring with unit such that $2 = 0$ in $R$, that is, $\text{char}(R) = 2$. Is there an example of such a ring with the map $x \mapsto x^2$ not being surjective on the invertible elements?

Answer 1

Sure, take $R = \mathbb{F}_2[[t]]$. The square of a unit $1 + a_1 t + a_2 t^2 + \dots$ has only terms of even degree $1 + a_1 t^2 + a_2 t^4 + \dots$. Actually I guess we can just take $R = \mathbb{F}_2[t]/t^2$ which should be a minimal counterexample (it is literally the smallest local ring of characteristic $2$ which is not a field).