When $p=2$, a non-unitary isometry on Hilbert space is unitarily equivalent to a unilateral shift so it cannot have Fredholm index zero.
When $p\in[1,\infty)\setminus\{2\}$, there is a version of the Wold decomposition theorem due to Faulkner-Huneycutt (Orthogonal decomposition of isometries in a Banach space) but I am unable to use it to draw any conclusion about the Fredholm index of a non-invertible isometry on $L^p$ space so I wonder if there is something known about this.