Southgate's men meet Croatia in the semi-finals on Wednesday after ending a 28-year wait just to reach the last four by cruising past Sweden in the quarter-finals. The ambulance used as a dance floor on Saturday was out of service.
In one video widely shared online, jubilant football fans can be seen jumping on furniture and kicking items around an IKEA store in Stratford, London following England's win against Sweden.
Police are aware of the damage to the vehicle, a London Ambulance Service spokesman confirmed.
Several people were pictured climbing on to the vehicle which was parked in Borough High Street while an officer responded to a call nearby.
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Online streaming service Spotify reports that Three Lions was streamed almost half a million times on England's match against Colombia, and has gone through the roof following the team's Sweden win.
But the London Ambulance Service reacted with disappointment to the celebrations, saying fans should be "better than this".
In Nottingham, a number of arrests were made for public order offences after a civilian auto was smashed and a group of men stood on a bus shelter, among other behaviour. The Sunday Express said a quietly confident nation has started to believe that football could really be "coming home", as England's 1996 soccer anthem goes.
"Being both British and Swedish, we were on the edge of our seats during the game and we would like to say 'grattis!' (that's congratulations in Swedish) to the England team for getting through to the semi-finals".