It depends on how one defines "starvation mode." The myth that you go into fat-store mode when low on calories is just that: a myth. The human body can't skip the laws of physics and allow you to be alive without using energy of some form. If you cut your intake down a ton, it means your body has less fuel to work with. You will 100% have side effects such as disorientation, lethargy and lack of energy, as well as possibly cognitive issues if you maintain an incredibly low calorie diet (referred to by some people who don't know what fasting is as "long term fasting") for any extended length of time. But at no point will you not be burning energy, and if you continue to eat at a caloric deficit you will continue to lose weight. The myth originates from the fact that if you are super sedentary, you burn less calories. Having little to no energy coming into your system will make you feel sort of like shit and probably not be nearly as active as normal, so your needs decrease along with the loss of activity. This can lead to slightly decreased weight loss when compared to the same caloric deficit and a higher level of activity.
However, starvation mode as in "you are literally starving" and your body starts burning everything to keep you physically alive is most definitely a thing. It's part of why keto diets are a thing. Your body -always- goes after sugars in the body first as an energy source. When it runs out or starts getting incredibly low, it starts using fats at a higher percentage. When you start getting low as a percentage of intake/bodymass in fats, your body starts consuming proteins. This can lead to degradation of your muscle mass and strength if kept at a very low level and sufficient protein isn't consumed to counterbalance it. It's why ketosis is considered an inherently catabolic diet, as once the fats calories are burned and someone isn't morbidly obese, your body starts synthesizing proteins at a higher rate, which is where the catabolic effect comes from. It's why people with active lifestyles tend to consume more protein than sedentary people while on ketogenic diets as a rule, because to keep a moderate/low protein intake would put their physical strength at risk.
But anyway. Starvation mode is used misleadingly by a lot of people in various ways. Not being able to lose weight while at a clear caloric deficit is most definitely a myth, however. The human body doesn't randomly skip thermodynamic laws when you are on a diet.