This paper proposes a global theoretically exact finite-time convergent observer for a class of nonlinear systems, which are not necessarily bounded-input-bounded-state with respect to the unknown inputs. With this aim, a scaled dissipative stabiliser is suggested ensuring that the highest derivative of the output estimation error is independent from the stabiliser gains. After this, a higher order sliding-mode differentiator with the gains adjusted to the upper bound of the unknown inputs is applied achieving finite-time convergence.