DFRobotshop Rover Shield power issue

Was running my rover when all of a sudden there was a small cloud of smoke from the rover shield. Switched it off, then on again and nothing was working.
Measurement between 5V and GND pins shows about 2.7V instead of 5V, which explains why nothing is working. Measurement at the battery connector terminals shows 3.9V (the battery is specified as 3.7V). If I connect the Arduino to USB while also supplying power from the battery, the rover is running, so either something is wrong with the battery or the battery connector, or the part of the circuit which is converting 3.7V to 5V is (partially) dead. Not sure which it is, but I’m hoping it’s the battery, since that is the cheapest part to replace. Any ideas?

Please confirm you are using a 3.7V LiPo which you purchased from RobotShop?

I used a 3.7V 1000mAh LiPo battery bought from Robotshop, yes. An issue I had with that battery, which may be related, was that the connector was a little loose, and a couple of times I had to push on the wires to get power.

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