At least the early albums in the Asterix series don't directly tell if the potion gives invincibility in addition to superhuman strength and speed. My impression has been that only Obelix is immortal due to falling to a potion cauldron as a child (the potion's effect on him is permanent). I think that invincibility would take away a lot of the tension from the stories and I doubt this has been René Goscinny's and Albert Uderzo's original intention.
For this answer, I considered all the books until Asterix in Belgium, i.e., all books with contributions from both, Uderzo and Goscinny. Since I only have the German editions, I won’t provide direct quotes. Page counts refer to the ones in the panels (usually at the bottom left) and should be independent of the edition.
A: with potion
This concerns the invulnerability of people that have drunk the potion (or can be assumed to have), except Obelix:
Asterix the Gaul, p. 11: The (yet unnamed) smith hammers a piece of metal with his bare fist.
Asterix the Gaul, p. 15: After drinking the potion, Caligula Minus hauls a bear-sized rock into the air and crushes himself with it. He quickly recovers. Getafix remarks that the potion does not make you invulnerable (for which an extra potion exists).
Asterix and the Goths, p. 8: In the druid show-off, the druid Valueaddedtax presents a potion that grants immunity to pain (and heat) while Getafix presents his potion. Valueaddedtax’ potion is not regarded as a subset of Getafix’ one.
Asterix and the Goths, p. 38: After Metric and Rhetoric have both drunk the potion and fought for hours, they are pretty battered up.
Asterix in Britain, p. 21: Obelix knocks out the legionary who drunk the potion (and a lot of wine) with one hit.
Asterix in Britain, p. 35: A knocked-out rugby player gets immediately revitalised when being fed the potion. This could be attributed to things like showmanship (pretending to be more injured than you are) and the placebo effect.
Asterix and the Laurel Wreath, p. 8: Asterix cautions that the potion does not grant invulnerability.
B: without potion
These are occasions where people who are not under the influence of the potion, are surprisingly robust:
Numerous occasions: Romans, pirates, Cacofonix, Goths, Normans, robbers, etc. survive and often even quickly recover from being beaten to pulp.
Numerous occasions: The pirates survive being shipwrecked at open sea.
Asterix the Gaul, p. 18: After the potion’s effect subsides, Caligula Minus gets crushed by another bear-sized rock and quickly recovers.
Asterix and Cleopatra, p. 24 ff.: Cleopatra’s taster survives quite some while after having eaten a blatantly poisoned cake (and then gets cured by Getafix).
Asterix and the Big Fight: Getafix, Psychoanalytix, and Cassius Ceramix survive being hit by a menhir. Psychoanalytix and Cassius Ceramix permanently lose their memory though.
Asterix and the Big Fight: Getafix survives various explosions.
Asterix in Switzerland: Vexatius Sinusitus survives being severely poisoned for at least a day without medical attention and for the entire adventure with Getafix’ medical attention, but without the antidote.
Asterix in Corsica, p. 19: The pirates survive their ship exploding.
Asterix and Caesar's Gift, p. 36: Asterix (who has not drunk the potion) literally catapults himself out of the Roman camp, suffering only a concussion.
Asterix in Belgium, p. 36: A Belgian survives a headshot by a catapult.
Numerous occasions: Obelix is unaffected by direct hits, e.g., in Asterix and the Roman Agent (p. 36) he ignores Magnumopus hitting his helmet-covered head with a club.
Numerous occasions: Obelix is incapacitated or knocked out by alcohol consumption.
Numerous occasions: Obelix eats a lot.
Asterix and the Banquet, p. 34: Obelix is unaffected by eating an entire boar poisoned with some sleeping drug. One bite of the same boar instantly knocks out Uptotrix.
Asterix and the Laurel Wreath, p. 19: Obelix is unaffected by the hangover cure from hell. He considers it stale.
Even without the potion, the Asterix universe is pretty lenient when it comes to dying and lasting damage (B1–10). There is possibly only one case where permanent damage is caused on page (B5).
The potion itself is not considered to grant invulnerability (A2, A3, A7) and this is also directly evidenced at times (A2, A4, A5). However, there are two exceptions where the potion boosts robustness (A1, A6). One of them (A1) is very early in the series and the other (A6) could have other reasons. In neither case, it grants clear invulnerabilty.
Even considering the leniency of the universe, Obelix is comparably resistant to mechanical damage (C1), but then again he is not that far out. He seems to have a peculiar digestive system (C2–C5), being immune to poisons (C3–4) other than alcohol (C2). The extents of this and whether this is connected to falling into the potion as a child remains unclear.
At least the early albums in the Asterix series don't directly tell if the potion gives invincibility in addition to superhuman strength and speed.
Actually, they do: the potion is often referred to as “the magic potion that makes one invincible”. I don't recall offhand if the word “invincible” (invincible) appears in the first album (Asterix the Gaul — Astérix le Gaulois). The text makes invincibility broadly synonymous with superhuman strength and speed.
Invincibility does not mean invulnerability. This is already established in the first album when a roman (Caligula Minus) drinks the potion (while disguised as a Gaul) and holds a boulder over his head. He throws the boulder in the air and it falls on his head, crushing him (with cartoon violence, as always in Asterix, so he gets out from under the boulder). Getafix (the druid) remarks that there is such a thing as a potion of invulnerability, but this is not the one. As far as I recall, a potion of invulnerability never comes up again.
Despite being a popular phrasing both in the books and in French popular culture, “magic potion of invincibility” is not quite accurate. It makes the drinker unbeatable in an unarmed fight (assuming the opponent hasn't drunk the potion himself: otherwise it's a draw, no matter how strong the fighters are naturally) — although the Normans (Asterix and the Normans) come close. But the potion does not make one immune to weapons such as spears and swords (for example, in Asterix and the Laurel Wreath (Les Lauriers de César), at one point, Astérix warns Obélix not to resist when they are confronted with a large troupe of elite Roman guards.). The Gaul village always beat the Roman legions around them in part because they mostly get the dregs: it's a posting that Roman soldiers dread, given out as punishment.
I think that invincibility would take away a lot of the tension from the stories and I doubt this has been René Goscinny's and Albert Uderzo's original intention.
It wasn't their original intention. But they knew of and embraced this effect when they made it a recurring element. In Goscinny raconte les secrets d'Astérix (René Goscinny, Cherche Midi, 2014) (my translation):
As for the magic potion that grants the indomitable Gauls superhuman strength, it was only meant to appear in the first episode. But it had so much success that we kept it. The potion (…) had unforeseeable consequences on the flow of the story. It killed suspense. We know in advance that the Gauls will win! But maybe this is an advantage. The reader never really fears for the Gauls… And never really fears the Romans… The magic potion is something I use to show that our wars are not serious, since all it takes to make everything better is a sip of a drink.