I don't believe so
While it's possible for language to develop without the concept of personal ownership (there's no concept or word for "that's mine"), I find it incredible that language could develop without the concept of "me."
Indeed, the only way I can imagine a creature without a sense of self is for that creature to have no reaction to its environment. Consider a bacteria or a virus. If the environment becomes adverse (say, pour alcohol onto the bacteria, killing it), the bacteria (insofar as I know, to be fair, I'm not a medical specialist) doesn't shy away from the threat of the alcohol. It just dies.
Compare this to a monkey, or a dog, or even a worm, where the creature reacts to its environment. If you poke at a worm, it tries to move away. Despite having no sapience at all, it is sentient enough to have a sense of "self."
Therefore, I believe it is impossible for communication (even in its most rudimentary form) to develop (i.e, "language") without the concept of self. The idea of "I don't want that to happen to me" always develops (insofar as we know from Earth evolution) before the ability to express the idea.
And just to make my point...
Let's assume a creature without a sense of self could speak. Let's ask it some questions.
No answer, the creature has no way of knowing if you're referring to yourself, itself, or another creature. Indeed, without a concept of self, it has no concept of of the Other, either. It stares at you, not knowing what you're referring to. Unless it's actually experiencing pain, then it might say "yes," but it's unlikely since it can't know who's experiencing the pain you're talking about.
No answer, the creature doesn't understand the pronouns "I" or "you" and so has no perspective concerning the point of reference. Frankly, without a sense of self, the creature has no way of thinking, "I'm in pain." All it can do is say, "Pain!" because, without a sense of self, it wouldn't understand the concept of "there is pain" because that embodies pain as an independent object. No sense of self, no independent objects, either. If the creature can comprehend the idea of "this rock, here," then it must comprehend "I" as well.
I think. This is a pretty alien point of view. But I don't believe it's possible. The capacity to discuss things would be so limited that it's hard to believe language could develop in the first place.