Lukashenko has not committed his small army to join Russia's war, but the risk of a new attack from Belarusian soil compels Ukraine to protect its northern border.
Russia has in recent weeks begun stationing tactical nuclear weapons in Belarus for the first time.
"Information obtained shows that during a training flight by US soldiers, contact with the drone was lost and it then fell into a forest area," the defense ministry wrote.
"Training or joint exercises of the Belarusian army and the Wagner Group is undoubtedly a provocation," Zbigniew Hoffmann told PAP.
"The armed forces of Belarus continue joint training with the fighters of the Wagner PMC (Private Military Company)," the Belarusian defense ministry said.
Despite his state of undress, Prigozhin has developed a reputation as a fashionista.
An adviser to Belarus' defence ministry said on Friday that nobody from the Wagner group had yet visited the disused military camp that Lukashenko had offered for the fighters' use.
"You know where you are situated geographically, and you know pretty well where the threat is coming from", said Lt Col Steffen Lieb, commander of the Patriot deployment.
Belarusian President Lukashenko stated he has a veto on use of Russian tactical nuclear weapons deployed on Belarusian territory.
Alexander Lukashenko, who has ruled Belarus since 1994, dismissed claims that he was "Europe's last dictator" on Thursday.