Context. Although the new generation of radial-velocity (RV) instruments such as ESPRESSO are expected to reach the long-term precision required to find other earths, the RV measurements are contaminated by some signal from stellar activity. This makes these detections hard. Aims. On real observations, we here demonstrate for the first time the effect of stellar activity on the RV of individual spectral lines. Recent studies have shown that this is probably the key for mitigating this perturbing signal. Results. We estimate that at least 89% of the lines that appear in the spectrum of alpha Cen B for which we measure a reliable RV are correlated with the stellar activity signal. This can be interpreted as those lines being sensitive to the inhibition of the convective blueshift observed in active regions. Because the velocity of the convective blueshift increases with physical depth inside the stellar atmosphere, we find that the effect induced by stellar activity on the RV of individual spectral lines is inversely proportional to the line depth. The stellar activity signal can be mitigated down to 0.8-0.9 m/s, either by selecting lines that are less sensitive to activity or by using the difference between the RV of the spectral lines that are formed at different depths in the stellar atmosphere as an activity proxy. Conclusions. This paper shows for the first time that based on real observations of solar-type stars, it is possible to measure the RV effect of stellar activity on the RV of individual spectral lines. Our results are very promising and demonstrate that analysing the RV of individual spectral lines is probably one of the solutions to mitigate stellar activity signal in RV measurements down to a level enabling the detection of other earths.