The 10, including two pregnant women, were allowed to disembark on the Mediterranean island of Lampedusa, which lies between the Italian mainland and the north African coast.
The others were three children, another woman, two sick men and two people accompanying them, media reported citing ministry sources. The vessel rescued 53 migrants drifting in an inflatable raft off the coast of Libya on Wednesday.
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On Thursday, Salvini, leader of the far-right League party, ordered Italian law enforcement to prevent the boat from entering Italian waters.
On Saturday, Salvini announced on Twitter: “I have just signed a ban on the entry, transit and berthing of Sea-Watch 3 in Italian territorial waters, as provided for by the Security Decree.” The decree still has to go before Italy's parliament where the coalition government holds a comfortable majority.
On Tuesday, Salvini said the 18-article decree would bring fines of up to 50,000 euros ($57,000) for the captain, owner and operator of a vessel “entering Italian territorial waters without authorisation”.
Salvini has seen his popularity soar in the last year after taking a hard line against migrants, which has included closing ports to rescue vessels.
Salvini says those setting sail from Libya to seek safety in Europe should be returned to the crisis-hit country — an order that is contentious under international law.
The NGO Sea Watch has repeatedly said it would not land survivors in Libya as it did not consider Tripoli a port of safety.
More than 12,000 people have died since 2014 trying to flee Libya to Europe by what the UN refugee agency calls the “world's deadliest sea crossing”.