Many networks generated by nature have two generic properties: they are formed in the process of {preferential attachment} and they are scale-free. Considering these features, by interfering with mechanism of the {preferential attachment}, we propose a generalisation of the Barabási--Albert model---the 'Fractional Preferential Attachment' (FPA) scale-free network model---that generates networks with time-independent degree distributions $p(k)\sim k^{-γ}$ with degree exponent $2<γ\leq3$ (where $γ=3$ corresponds to the typical value of the BA model). In the FPA model, the element controlling the network properties is the $f$ parameter, where $f \in (0,1\rangle$. Depending on the different values of $f$ parameter, we study the statistical properties of the numerically generated networks. We investigate the topological properties of FPA networks such as degree distribution, degree correlation (network assortativity), clustering coefficient, average node degree, network diameter, average shortest path length and features of fractality. We compare the obtained values with the results for various synthetic and real-world networks. It is found that, depending on $f$, the FPA model generates networks with parameters similar to the real-world networks. Furthermore, it is shown that $f$ parameter has a significant impact on, among others, degree distribution and degree correlation of generated networks. Therefore, the FPA scale-free network model can be an interesting alternative to existing network models. In addition, it turns out that, regardless of the value of $f$, FPA networks are not fractal.