  We prove a general existence and uniqueness theorem for holomorphic
perturbative solutions of singularly perturbed nonlinear complex differential
systems of the form $\hbar \partial_x f = F (x, \hbar, f)$ where $\hbar$ is a
small complex perturbation parameter. Specifically, we give conditions that
allow us to promote, in a unique and precise sense, a formal perturbative
solution $\hat{f} (x, \hbar)$ to a holomorphic solution $f (x, \hbar)$ whose
perturbative expansion is $\hat{f}$. Furthermore, we prove that $f$ is the
uniform Borel resummation of $\hat{f}$. As sample applications, we analyse the
Painlev\'e I equation as well as derive exact WKB solutions for a generalised
3\textsuperscript{rd}-order Airy equation.
