SIM800L occasionally hitting battery back with 0.3 ms voltage spikes

I have experimented a bit with SIM800L and wired it up together with Arduino. I managed to get it working (sending HTTP requests every couple of seconds etc.). I have replaced the default cap from the Chinese board with a 470 uF one because it did not work stably for me before that.

I’m powering the module with a li-ion battery, and I had some issues that I thought could be caused by the battery, so I have looked at the battery voltage under the oscilloscope. The voltage occassionally drops from 3.8 to 3.6V, and after that some voltage spikes can be seen (probably when the module is transmitting something to the network). I suspect it might be induced voltage from a coil (?), however the thing that puzzles me is how quickly the voltage drops after going high - it seems like an almost perfect square signal. I attached two oscilloscope graphs (one recorded within a higher timeframe).

This all also occurs when SIM800L is not connected to anything else but the battery.

Could anyone suggest what the cause of this could be?

I would expect to see what you are, the unit draws 2 Amps Max when transmitting. Since the battery is not a zero impedance device I would expect some voltage drop when the transmitter is on. My rough calculation shows the battery has a 0.1 Ohm impedance. Read the SIM800L data sheet for more information and about using a 4.7 uF bypass capacitor. This response is to help you get started in solving your problem, not solve it for you.
Good Luck & Have Fun!
Gil