Is there any reason that we couldn't believe in life on another planet somewhere? Is there any source that indicates to the contrary?
In this article, Rabbi Aryeh Kaplan sources and discusses opinions on both sides of the argument, bringing various proofs and questions.
Chasdai Crescas says aliens can exist, Yosef Albo said there cannot be any other beings with free will, apparently the only objection with believing in aliens centers around this issue of free will. Rabbi Kaplan also quotes the Sefer HaBris who took the middle road saying they exist, but do not have free will. Rabbi Kaplan goes on to say that the Zohar seems to support belief in extra terrestrial life.
The Sefer Hikrim brings his rebbe which held that there are no aliens. However the Sefer Hikrim himself argues and holds that there are aliens. He brings a proof from the possuk in Shoftim which says "oro mroz", which the gamera in Moed Katan (Rashi brings this) says roz is a star, and the passuk concludes "arur yoshveha" cursed are its inhabitants, clearly there is some form of life in space.
User6591 answered based on the essay entitled "On Extraterrestrial Life" by Rabbi Kaplan that there are mutiple opinions. That essay is written in the same author's book, Moreh Ohr on pp. 47-50. All of the relevant information is laid out and presented with the relevant passages quoted in their original. He then concludes with this:
We see from this [starting on p. 47] that there is a singular species in the world that is capable of free will, Torah, reward and punishment which are the purpose of creation. This is mankind, to whom God has given The One Torah. It is however possible that there exists many species of living creatures on other stars i.e. planets, which may even be capable of intelligence and wisdom, just not freewill. The subject of freewill is but a very small item as it is, that cannot be verified in science [or by scientists], only by belief in our Torah. And if they do find additional species of living creatures, we have already learned from the Torah that don't have freewill.
According to my understanding of the Torah, and my logic:
Torah is one and unchangeable
Only Adam was created בצלמנו כדמותנו and יודע טוב ורע and only his descendants can qualify as such.
While we didn't see men leaving Earth for other planets, there could be no human-like, sentient life besides humans.
If you asked such a question to Rambam or the Geonim or the Sages that would be definitely considered heresy as the Earth was flat and there was a clear distinction between the Earth and Heavens (see Rambam's Yesodey Hatorah - everything he spoke of was physical to him and his generation). All of the sudden אנן סהדי that we can fly into the skies and there's no רקיע - firmament whatsoever, so we immediately rewrite Judaism to accommodate for that apparent knowledge.
Therefore the most problematic part in this question - what do we call Judaism? How far can we deviate from our Rabbis' understanding of the Torah to fit it into contemporary science? I really don't know what the limits are.
Rav Meir Mazuz has a very interesting article on it which he claims there are no life on other planets as on earth.
The torah's word is above everything, above all the professor's, above all the scientists. All their words are null and void against one word of torah!
For example: "in the beginning, G-d created the heavens and the earth" (Gen.1:1). The deniers came along and said: This is false! The earth is part of the heaven. We once had a book called "the heavens" (by the french astronomer, K.Palmerion). He wrote there: "the heavens are everything. Don't think the heavens are one thing and the earth is another. Rather, the earth is just one planet in the heaven!" He was not the first to say this. After it was discovered that all the planets revolve around the sun, then all the scientists also held this view.
Thus they said, what is this: "the heavens and the earth!?" (Gen 1:1)
When they reached this conclusion, they asked themselves: "just like there are human beings on the earth, so too there must be on other planets. There must exist also on the moon, on mars, on venus, mercury, etc.
In truth, there were some torah sages who thought like this. In the days of the Chacham Tzvi, there was a Rav talmid chacham, tzadik, G-d fearing, by the name of David Nito, author of the book Matei Dan. There they asked him: "what do you think of what the scientists say that there are human beings on the moon?". He answered: "Why not? If they tell you there are human beings in India, will you say they walk upside down or they have three eyes?! Of course not. All are the same. So too here. All the planets are the same".
Eventually, this matter was proven false. When? About fifty years ago, when they went up on the moon and saw no men, no horses, etc. There's no air and no water. All desolate wasteland (at that time, Rabbi Nissan Pinson told me: "slowly, slowly they will realize that there is no place fit for life like planet earth".)
For the "heavens" is a vast sphere which has nothing, while the "earth" has so many interesting things. There are human beings, animals, trees, grasses, and there are people who recognize their Creator, pray to Him and thank Him.
This is what is written: "in the beginning, G-d created the heavens and the earth". It has already been fifty years, since the scientists search against this. But in truth, there is nothing. Period.
One word of torah is worth more than all the scientists' words together! They are all as dust and ashes against it. So too, for everything else written in the torah. (from Rabbi Mazuz's weekly class. Parsha Tzav 5778) note: this phenomena is happening also in physics, biology, cosmology, etc. For example, the scientists expected living cells to be simple but even the simplest bacteria turned out to be an intense world of bewildering complexity. Likewise for everything. Even the tiny atom is an intense world of mathematics which gets more complex the more you go in.