That's not how hits work.
Other answers here have ably described that this is not how lances work, RAW, in 5e.
But there is a more fundamental misunderstanding here - that of how combat works.
As described, the OP's DM has a system in which lance "hits"1 always pierce armor and impale the target, while "misses" sometimes narratively strike but are stopped by armor, and sometimes miss entirely. This confuses the game term "hit" with not only a narrative impact on armor, but one that pierces the armor and does grievous and visible bodily damage. This is simply not how combat works.
Actually, the rules for making an attack state:
- Resolve the attack. You make the attack roll. On a hit, you roll damage, unless the particular attack has rules that specify otherwise.
The rules for making a "hit" state: if you "hit", you roll damage. That's it and that's all. Nothing there says you necessarily strike your opponent in a narrative sense, and nothing says you must certainly pierce their armor and impale them. You most likely reduce their hp; that is the only rules-required consequence.
What are hit points?
Hit points represent a combination of physical and mental durability, the will to live, and luck...Whenever a creature takes damage, that damage is subtracted from its hit points. The loss of hit points has no effect on a creature's capabilities until the creature drops to 0 hit points.
A successful "hit" reduces the target's hit points, which are some combination of its durability, will, and luck. It is possible for a successful "hit" to be a narrative miss, since it doesn't result in any apparent physical damage or reduction in their immediate effectiveness. This is sometimes called 'getting stabbed in the luck'. In fact, the PHB suggests that while you are above half max hp, "hits" don't have any visible narrative effects. The "Describing the Effects of Damage" sidebar (p. 197) says:
Dungeon Masters describe hit point loss in different ways. When your current hit point total is half or more of your hit point maximum, you typically show no signs of injury. When you drop below half your hit point maximum, you show signs of wear, such as cuts and bruises.
So while OP's DM is empowered to describe their combat narrative the way they would like, it would be passing strange to have a lance "hit" against an armored foe narratively pierce the armor and impale them if they were left with more than half their hp. More likely a "miss" would be a narrative miss, while a "hit" that left them with many hp would be a narrative near miss that damaged their luck but did not impact the lance. A "hit" that left them approaching half hp would be a narrative hit that was deflected by their armor, and only "hits" that left them with fewer than half their hp would cause narrative wounds, which would be 'cuts and bruises' rather than impalement. Impaling would be reserved for the consequence of a hit that brought them very close to 0hp. Which brings us too...
Knocking a creature out:
Sometimes an attacker wants to incapacitate a foe, rather than deal a killing blow. When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out. The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable.
If impaling a creature with a lance is a precursor to killing it in combat, the attacking player gets to make that choice, not the DM. The lance is a melee weapon, and OP should be able to choose to knock foes unconscious rather than kill them. The DM might choose to describe that as the lance shattering on the foe's helmet as it is dealt the knock-out blow, but it is hard to imagine how impaling someone with a lance would leave them stable and unconscious.
As other answers have indicated, the DM's combat system is one that is specifically designed to make lances single-use weapons, either through their destruction or having them remain stuck in targets. However, it does so through a fundamental change in the combat rules. For OP's DM, a "hit" is always a narrative hit that buries the weapon in the foe, while a "miss" is a narrative hit half the time as well. In the RAW combat system, all "misses" are clean narrative misses, and even "hits" are sometimes narrative near misses and sometimes glancing blows depending on the target's hp total. It is the DM's prerogative to alter the combat system thusly, but OP should approach their conversation with their DM armed with the knowledge of just how radical a departure from RAW this is.
Does piercing damage ignore Armor Class?
How can I describe hit point damage without talking about wounds?
How to flavor my narration of a fight in a creatively plausible way without affecting mechanics?
How should I handle players wanting to aim their attacks at specific body parts?
1 Herein I use "hit" and "miss" in quotations for the game-defined meaning of both, and hit and miss without quotations for the narrative event.