Suppose that $R$ is a ring, not necessarily commutative nor associative. Assume that for every $a \in R$, the left multiplication map $$ \lambda_a \colon R \to R \colon x \mapsto ax $$ is invertible. (We do not assume that its inverse is again a left multiplication map $\lambda_b$ for some $b \in R$.)

Is such a ring $R$ necessarily unital (i.e., does it have a unit $1 \in R$)?