We present the first attempt to detect outflows from galaxies approaching the Epoch of Reionization (EoR) using a sample of nine star-forming (SFR = 31 +/- 20M(circle dot) yr(-1)) z similar to 5.5 galaxies for which the [C II] 158 mu m line has been previously obtained with Atacama Large Millimeter Array (ALMA). We first fit each line with a Gaussian function and compute the residuals by subtracting the best-fitting model from the data. We combine the residuals of all sample galaxies and find that the total signal is characterized by a flux excess of similar to 0.5 mJy extended over similar to 1000 km s(-1). Although we cannot exclude that part of this signal is due to emission from faint satellite galaxies, we show that the most probable explanation for the detected flux excess is the presence of broad wings in the [C II] lines, signatures of starburst-driven outflows. We infer an average outflow rate of. M(over dot) = 54 +/- 23M(circle dot) yr(-1), providing a loading factor eta = M(over dot)/SFR = 1.7 +/- 1.3 in agreement with observed local starbursts. Our interpretation is consistent with outcomes from zoomed hydrosimulations of Dahlia, a z similar to 6 galaxy (SFR similar to 100M(circle dot) yr(-1)), whose feedback-regulated star formation results into an outflow rate. M(over dot) similar to 30M(circle dot) yr(-1). The quality of the ALMA data is not sufficient for a detailed analysis of the [C II] line profile in individual galaxies. Nevertheless, our results suggest that starburst-driven outflows are in place in the EoR and provide useful indications for future ALMA campaigns. Deeper observations of the [C II] line in this sample are required to better characterize feedback at high-z and to understand the role of outflows in shaping early galaxy formation.