While stretching of a polymer along a flat surface is hardly different from the classical Pincus problem of pulling chain ends in free space, the role of curved geometry in conformational statistics of the stretched chain is an exciting open question. Here by means of the scaling analyses and computer simulations we examine stretching of a fractal polymer chain around a disc in 2D (or a cylinder in 3D) of radius $R$. Surprisingly, we reveal that the typical excursions of the polymer away from the surface scale as $\Delta \sim R^{\beta}$, with the Kardar-Parisi-Zhang (KPZ) growth exponent $\beta=1/3$, for any fractal dimension of the chain. Moreover, we find that the curvature-induced correlation length of a fractal chain behaves as $S^* \sim R^{1/z}$ with the KPZ dynamic exponent $z=3/2$, suggesting that the crossover from flat to curved geometry of a stretched polymer corresponds to the crossover from large to short time scales in the KPZ stochastic growth. Thus, we argue that curvature of an underlying boundary furnishes universal KPZ-like statistics to the stretched fractal paths, which further suggests numerous connections with several branches of mathematical physics.