We consider the problem of estimating the states of a continuous-time nonlinear dynamical system, where a subset of sensors can be maliciously corrupted using a potentially unbounded additive signal. The proposed estimation scheme employs a bank of observers which are robust with respect to disturbances and attacks, in conjunction with median operation to build the state estimate. The median operation is the key ingredient which guarantees that the state estimate is constructed using sensor(s) which are not under attack. A standing assumption in this scheme is that the system has to be observable from each sensor. We provide a constructive design method for the state observers for a class of nonlinear systems and illustrate the efficacy of the resilient median-based state estimation scheme using real data on an inverter-based power distribution network.