user-1: Do Buddhists claim that if there was a self it could not be annihilated? I dimly remember reading this, but no argument or anything like that.

user471: In addition to the answer by ChrisW, the following sutta quotes provide further perspective on the same topic.
From SN 22.93:

At Savatthi. “Bhikkhus, suppose there was a mountain river sweeping
downwards, flowing into the distance with a swift current. If on
either bank of the river kasa grass or kusa grass were to grow, it
would overhang it; if rushes, reeds, or trees were to grow, they would
overhang it. If a man being carried along by the current should grasp
the kasa grass, it would break off and he would thereby meet with
calamity and disaster; if he should grasp the kusa grass, it would
break off and he would thereby meet with calamity and disaster; if he
should grasp the rushes, reeds, or trees, they would break off and he
would thereby meet with calamity and disaster.
“So too, bhikkhus, the uninstructed worldling … regards form as self,
or self as possessing form, or form as in self, or self as in form.
That form of his disintegrates and he thereby meets with calamity and
disaster. He regards feeling as self … perception as self … volitional
formations as self … consciousness as self, or self as possessing
consciousness, or consciousness as in self, or self as in
consciousness. That consciousness of his disintegrates and he thereby
meets with calamity and disaster.
“What do you think, bhikkhus, is form permanent or
impermanent?”—“Impermanent, venerable sir.”…—“Therefore … Seeing thus
… He understands: ‘… there is no more for this state of being.’”

From MN 22:

“Venerable sir, can there be agitation about what is non-existent
internally?”
“There can be, bhikkhu,” the Blessed One said. “Here, bhikkhu, someone
has the view: ‘That which is the self is the world; after death I
shall be permanent, everlasting, eternal, not subject to change; I
shall endure as long as eternity.’ He hears the Tathāgata or a
disciple of the Tathāgata teaching the Dhamma for the elimination of
all standpoints, decisions, obsessions, adherences, and underlying
tendencies, for the stilling of all formations, for the relinquishing
of all attachments, for the destruction of craving, for dispassion,
for cessation, for Nibbāna. He thinks thus: ‘So I shall be
annihilated! So I shall perish! So I shall be no more!’ Then he
sorrows, grieves, and laments, he weeps beating his breast and becomes
distraught. That is how there is agitation about what is non-existent
internally.”
“Venerable sir, can there be no agitation about what is non-existent
internally?”
“There can be, bhikkhu,” the Blessed One said. “Here, bhikkhu, someone
does not have the view: ‘That which is the self is the world…I shall
endure as long as eternity.’ He hears the Tathāgata or a disciple of
the Tathāgata teaching the Dhamma for the elimination of all
standpoints, decisions, obsessions, adherences, and underlying
tendencies, for the stilling of all formations, for the relinquishing
of all attachments, for the destruction of craving, for dispassion,
for cessation, for Nibbāna. He does not think thus: ‘So I shall be
annihilated! So I shall perish! So I shall be no more!’ Then he does
not sorrow, grieve, and lament, he does not weep beating his breast
and become distraught. That is how there is no agitation about what is
non-existent internally.


user254: The first and perhaps most famous sutta like that which comes to mind is the Anatta-lakkhana Sutta: The Discourse on the Not-self Characteristic (SN22.59) translated here and here.
I think that says that if something (e.g. form or consciousness or feeling or perception) is impermanent, then it's "not fit" to be regarded as "self".
The word translated as "not fit" is kalla ("clever", "healthy", etc.).

-1: <p>Do Buddhists claim that if there was a self it could not be annihilated? I dimly remember reading this, but no argument or anything like that.</p>
 471: <p>In addition to the answer by ChrisW, the following sutta quotes provide further perspective on the same topic.</p>
<p>From <a href="https://suttacentral.net/sn22.93/en/bodhi" rel="nofollow noreferrer">SN 22.93</a>:</p>
<blockquote>
<p>At Savatthi. “Bhikkhus, suppose there was a mountain river sweeping
downwards, flowing into the distance with a swift current. If on
either bank of the river kasa grass or kusa grass were to grow, it
would overhang it; if rushes, reeds, or trees were to grow, they would
overhang it. If a man being carried along by the current should grasp
the kasa grass, it would break off and he would thereby meet with
calamity and disaster; if he should grasp the kusa grass, it would
break off and he would thereby meet with calamity and disaster; if he
should grasp the rushes, reeds, or trees, they would break off and he
would thereby meet with calamity and disaster.</p>
<p>“So too, bhikkhus, the uninstructed worldling … regards form as self,
or self as possessing form, or form as in self, or self as in form.
That form of his disintegrates and he thereby meets with calamity and
disaster. He regards feeling as self … perception as self … volitional
formations as self … consciousness as self, or self as possessing
consciousness, or consciousness as in self, or self as in
consciousness. That consciousness of his disintegrates and he thereby
meets with calamity and disaster.</p>
<p>“What do you think, bhikkhus, is form permanent or
impermanent?”—“Impermanent, venerable sir.”…—“Therefore … Seeing thus
… He understands: ‘… there is no more for this state of being.’”</p>
</blockquote>
<p>From <a href="https://suttacentral.net/mn22/en/bodhi" rel="nofollow noreferrer">MN 22</a>:</p>
<blockquote>
<p>“Venerable sir, can there be agitation about what is non-existent
internally?”</p>
<p>“There can be, bhikkhu,” the Blessed One said. “Here, bhikkhu, someone
has the view: ‘That which is the self is the world; after death I
shall be permanent, everlasting, eternal, not subject to change; I
shall endure as long as eternity.’ He hears the Tathāgata or a
disciple of the Tathāgata teaching the Dhamma for the elimination of
all standpoints, decisions, obsessions, adherences, and underlying
tendencies, for the stilling of all formations, for the relinquishing
of all attachments, for the destruction of craving, for dispassion,
for cessation, for Nibbāna. He thinks thus: ‘So I shall be
annihilated! So I shall perish! So I shall be no more!’ Then he
sorrows, grieves, and laments, he weeps beating his breast and becomes
distraught. That is how there is agitation about what is non-existent
internally.”</p>
<p>“Venerable sir, can there be no agitation about what is non-existent
internally?”</p>
<p>“There can be, bhikkhu,” the Blessed One said. “Here, bhikkhu, someone
does not have the view: ‘That which is the self is the world…I shall
endure as long as eternity.’ He hears the Tathāgata or a disciple of
the Tathāgata teaching the Dhamma for the elimination of all
standpoints, decisions, obsessions, adherences, and underlying
tendencies, for the stilling of all formations, for the relinquishing
of all attachments, for the destruction of craving, for dispassion,
for cessation, for Nibbāna. He does not think thus: ‘So I shall be
annihilated! So I shall perish! So I shall be no more!’ Then he does
not sorrow, grieve, and lament, he does not weep beating his breast
and become distraught. That is how there is no agitation about what is
non-existent internally.</p>
</blockquote>
 254: <p>The first and perhaps most famous sutta like that which comes to mind is the <em>Anatta-lakkhana Sutta: The Discourse on the Not-self Characteristic</em> (SN22.59) translated <a href="https://suttacentral.net/sn22.59/en/sujato" rel="nofollow noreferrer">here</a> and <a href="https://www.accesstoinsight.org/tipitaka/sn/sn22/sn22.059.nymo.html" rel="nofollow noreferrer">here</a>.</p>
<p>I think that says that if something (e.g. form or consciousness or feeling or perception) is impermanent, then it's &quot;not fit&quot; to be regarded as &quot;self&quot;.</p>
<p>The word translated as &quot;not fit&quot; is <a href="https://suttacentral.net/define/kalla" rel="nofollow noreferrer">kalla</a> (&quot;clever&quot;, &quot;healthy&quot;, etc.).</p>
