X-rays have been observed in natural downward cloud-to-ground lightning for over 20 years and in rocket-triggered lightning for slightly less. In both cases, this energetic radiation has been detected during the stepped and dart leader phases of downward negative flashes. More recently, X-rays have also been reported during the dart leader phase of upward negative flashes. In this study, we present the observations of four upward positive lightning flashes from the Säntis Tower (2.5 km ASL) in Switzerland. These consist of the simultaneous records of electric current passing through the tower, and electric field strength and X-ray flux 20 m from the tower base. One of the flashes was captured by a high-speed camera operating at 24,000 frames per second, stills from which are also presented. We detected X-rays during the initial phase of upward negative leader propagation, which can be associated with the leader-stepping process from electric field and current waveforms. To the best of our knowledge, this is the first time that such measurements are reported in the literature. The obtained time-synchronised data confirm that the X-ray emissions detected are associated with the initial steps of the upward negative leader. The frequency and energy of X-ray pulses appear to decrease as functions of time, with pulses disappearing altogether within the first millisecond of the leader initiation. X-ray emission also appears to be correlated with the maximum current-derivative and the electric field change of leader steps, consistent with cold electron runaway. These observations contribute to improving our understanding of upward lightning, which is a primary source of damage to tall structures such as wind turbines and telecommunications towers, as well as aircraft during takeoff and landing.