Using high-resolution collisionless N-body simulations, we study the properties of tidal tails formed in the immediate vicinity of a two-component dwarf galaxy evolving in a static potential of the Milky Way (MW). The stellar component of the dwarf is initially in the form of a disc and the galaxy is placed on an eccentric orbit motivated by cold dark matter based cosmological simulations. We measure the orientation, density and velocity distribution of the stars in the tails. Due to the geometry of the orbit, in the vicinity of the dwarf, where the tails are densest and therefore most likely to be detectable, they are typically oriented towards the MW and not along the orbit. We report on an interesting phenomenon of `tidal tail flipping': on the way from the pericentre to the apocentre, the old tails following the orbit are dissolved and new ones pointing towards the MW are formed over a short time-scale. We also find a tight linear relation between the velocity of stars in the tidal tails and their distance from the dwarf. Using mock data sets, we demonstrate that if dwarf spheroidal (dSph) galaxies in the vicinity of the MW are tidally affected their kinematic samples are very likely contaminated by tidally stripped stars which tend to artificially inflate the measured velocity dispersion. The effect is stronger for dwarfs on their way from the pericentre to the apocentre due to the formation of new tidal tails after pericentre. Realistic mass estimates of dSph galaxies thus require removal of these stars from kinematic samples.