We establish the first existence and uniqueness result for mild solutions of abstract stochastic evolution equations driven by arbitrary cylindrical Lévy processes in Hilbert spaces. The coefficients are assumed to satisfy global Lipschitz conditions, and no moment assumptions are imposed on the driving noise. The principal difficulty arises from the fact that cylindrical Lévy processes exist solely in a generalised sense and typically admit no semimartingale or Lévy-Itô decomposition, which precludes the use of classical existence methods. To overcome these obstacles, we develop a pathwise adaptive Euler-Peano approximation scheme based on noise-dependent stopping times and a fixed-point formulation of the mild solution operator. The resulting approach avoids stochastic calculus techniques relying on semimartingale decompositions and provides a robust and flexible framework for treating multiplicative cylindrical Lévy noise in infinite-dimensional systems.