Let {chi(k)(t), t >= 0} be a stationary chi-process with k degrees of freedom being independent of some non-negative random variable T. In this paper we derive the exact asymptotics of P {sup(t is an element of [0, T]) chi(k) (t) > u} as u -> infinity when T has a regularly varying tail with index lambda is an element of [0, 1). Three other novel results of this contribution are the mixed Gumbel limit law of the normalised maximum over an increasing random interval, the Piterbarg inequality and the Seleznjev pth-mean theorem for stationary chi-processes.