  We study the problem where a one-dimensional elastic string is immersed in a
two-dimensional steady Stokes fluid. This is known as the Stokes immersed
boundary problem and also as the Peskin problem. We consider the case with
equal viscosities and with a fully non-linear tension law; this model has been
called the fully nonlinear Peskin problem. In this case we prove local in time
well-posedness for arbitrary initial data in the scaling critical Besov space
$\dot{B}^{3/2}_{2,1}(\mathbb{T} ; \mathbb{R}^2)$. We additionally prove the
optimal higher order smoothing effects for the solution. To prove this result
we derive a new formulation of the boundary integral equation that describes
the parametrization of the string, and we crucially utilize a new cancellation
structure.
