It’s possible. Your typical surge protector will have things like a choke (just a coil of wire), a fuse and/or breaker, and some metal oxide varistors (MOVs). A surge could have punched through the insulation in the choke and it could be shorting out and making smoke now.
MOVs work by turning “on” when the voltage gets too high. The MOV then essentially creates a short circuit between the protected line and ground. By creating an intentional short circuit fault, the MOV clamps the incoming surge and shunts it to ground, but the MOV can only handle so much power before it gets completely fried. The higher the joule rating on your power strip, the more power the MOVs can handle before they get toasted.
Due to the way MOVs work, they are usually destroyed in the process of protecting a surge. If you are handy with electronics you can replace the MOVs yourself. Normal folks just buy a new surge protector.
MOVs may make smoke when they first blow, but I wouldn’t expect them to keep making smoke when you plug the power strip in later. Either you’ve got a shorted choke or you’ve got some fancier circuit in there that’s been damaged, or maybe you’ve still got some problems in your wiring.
This makes me think that they basically lost the neutral connection. With a floating neutral, you still get 240 volts from line to line, but your line to neutrals will be proportional to the current flowing between the two circuits. So if twice as much current is flowing through one than the other you’ll get 80 volts on one circuit and 160 volts on the other, for example, and those voltage values will change as the current being drawn changes. The overvoltage can cause excessive current to flow which can then cause further damage to the electrical system, to you can easily end up with much more than just the original broken neutral problem.
Surge protectors don’t do diddley to protect you from brownouts, and they won’t protect you from an overvoltage until the voltage gets up high enough to trigger them. Surge protectors are designed for short high voltage surges. Many of them don’t offer much protection against a moderate overvoltage like what you get from a floating neutral.
In a good surge protector, the LEDs also won’t light if the MOVs are blown. It gives you a good visual indication as to whether or not you still have good surge protection.
With your typical el-cheapo surge protector, you have no way of knowing if the MOVs already took a hit and are blown or not.