A third victim of a Ghanaian man who allegedly went on the rampage with a pickaxe in Milan this week died of his wounds on Tuesday in a case that has inflamed the debate over undocumented migrants.

Mada Adam Kabobo, a homeless 21-year-old, was arrested on Saturday after an apparently random early morning spree in Italy's commercial capital.

Grainy closed-circuit footage broadcast on Italian television appeared to show Kabobo roaming the streets with the pickaxe slung over his shoulder.

The third victim was a 64-year-old pensioner walking his dog. The first two were a 21-year-old delivering newspapers and an unemployed man, 40.

Kabobo arrived in Italy illegally in 2011 and his request for asylum had been refused, but he had not been deported since he had appealed the ruling.

He had been arrested several times for alleged offences including theft and vandalism.

He has told investigators in prison that he heard voices in his head, Italian media reported.

Immigration is a sensitive topic in Italy. Silvio Berlusconi's People of Freedom party has said the army should be deployed to improve security and far-right parties say the case shows that undocumented migrants should be expelled.

Four unlit Molotov cocktails were found outside a refugee centre near the scene of the deadly rampage on Monday.

Tens of thousands of African migrants have landed on Italian shores in recent years and rights groups have called for them to better integrated instead of being forced to live in the shadows.