for a ring $R$ with unity , let $U(R)$ denote the group of units of $R$ . Now there are lots of finite commutative rings, of arbitrarily high order, with exactly one unit ; indeed $U(R)=1$ for a finite commutative ring $R$ iff $a^2=a , \forall a \in R$ . Incidentally , I couldn't find any finite non-commutative ring with exactly one unit ; matrix rings doesn't seem to work.

So my question is : Does there exist a finite non-commutative ring with unity having exactly one invertible (unit) element ?

Small remark : Note that such a ring must have characteristic $2$