The shop targeted in Sloane Square is Gucci's flagship store.
It's thought that the Mercedes used to reverse into the glass doors may have been stolen.
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A woman died as she crossed the road when she was struck by four separate motorists who police believe did not stop.
Patrols including local volunteers have been set up after the grisly finds in Thamesmead in the last month.
Thieves targeted a property in Richmond on Monday afternoon.