My question: Is the left invertible element unique in Banach algebras? I think it is not true. But it is not easy for me to find a good example.
A simple example of left invertible operator in Banach algebra $\mathcal{B}(l^2(\mathbb{N}))$ is unilaterial right shift $$T_1(x_1,x_2,x_3,...)=(0,x_1,x_2,...).$$ However its left inverse is unique.
If the operator $T$ is left invertible its range is closed as for $ST=I$ we have $$\|x\|=\|STx\|\le \|S\|\,\|Tx\|$$ For any left invertible operator $T:X\to X,$ such that $T(X) \subsetneq X,$ the left inverse is not unique. Indeed, there exists a bounded linear functional $\varphi\neq 0$ satisfying $T(X)\subset \ker \varphi$ (this holds for any proper closed subspace of $X$, as a consequence of the Hahn-Banach theorem). Fix a nonzero element $x_0\in X$ and a number $a.$ Let $S$ be a left inverse of $T.$ Then the operator $S_a(x)=Sx+a\,\varphi(x)x_0$ is a left inverse as well, since $$S_a(Tx)=STx+a\,\varphi(Tx)x_0=STx=x$$