Intelligent reflecting surface (IRS) is a promising technology for wireless communications, thanks to its potential capability to engineer the radio environment. However, in practice, such an envisaged benefit is attainable only when the passive IRS is of a sufficiently large size, for which the conventional uniform plane wave (UPW)-based channel model may become inaccurate. In this paper, we pursue a new channel modelling and performance analysis for wireless communications with extremely large-scale IRS (XL-IRS). By taking into account the variations in signal's amplitude and projected aperture across different reflecting elements, we derive both lower- and upper-bounds of the received signal-to-noise ratio (SNR) for the general uniform planar array (UPA)-based XL-IRS. Our results reveal that, instead of scaling quadratically with the increased number of reflecting elements M as in the conventional UPW model, the SNR under the more practically applicable non-UPW model increases with M only with a diminishing return and gets saturated eventually. To gain more insights, we further study the special case of uniform linear array (ULA)-based XL-IRS, for which a closed-form SNR expression in terms of the IRS size and transmitter/receiver location is derived. This result shows that the SNR mainly depends on the two geometric angles formed by the transmitter/receiver locations with the IRS, as well as the boundary points of the IRS. Numerical results validate our analysis and demonstrate the importance of proper channel modelling for wireless communications aided by XL-IRS.