Žampa comes from Starý Smokovec in the High Tatras . In December 2005, at the age of 15, he competed in his first FIS races . Notable successes were initially lacking in the following years. Without ever having started in the European Cup or the World Cup , he took part in the 2009 World Cup in Val-d'Isère and finished 21st in the slalom. This was followed by participation in the Junior World Championships 2009 in Garmisch-Partenkirchen , where he finished 12th in the combination . He made his debut in the European Cup in December 2009. In February 2010 he won the Slovak Junior Slalom Championship and his first FIS race, in March 2010 the Slovak Giant Slalom Championship.
On November 14, 2010, Žampa started the World Cup for the first time and was eliminated in the first round of the slalom in Levi . At the 2011 World Championships in Garmisch-Partenkirchen, he finished 24th in slalom. In the 2011/12 season he began to classify himself regularly in the points in the European Cup. On January 23, 2012, he achieved third place in the giant slalom in Zell am See , the best classification so far. At the end of the season he won two more Slovak championship titles. In preparation for the 2012/13 season, Žampa competed in numerous races in Oceania in the summer; he got nine wins (eight in a row). He scored his first World Cup points on October 28, 2012 with ninth place in the giant slalom in Sölden , with the second-best time in the second run. In the course of the rest of the season he did not get beyond a 16th place.