If you want to handle missing files, don't use the diamond, use open or something like

BEGIN { -f or die "No file $_" for @ARGV }
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BTW, $! only contains a meaningful value right after something went wrong. If you print it after processing an existing file, its value might be totally irrelevant (I'm getting Inappropriate ioctl for device).

However, after the loop, there doesn't seem to be a way for the program to detect that a file wasn't processed. In real life, that could be awkward.

Simple solution: use strictures.

perl -Mstrictures -e'while(<>){1;}continue{print "$ARGV\n" if eof;}; p
+rint $!' file1 file2 file3
file1
Can't open file2: No such file or directory at -e line 1, <> line 1.
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That should make it pretty obvious even to the most obtuse of users that something has gone badly wrong.

If you are after a less abrupt, more hand-holding approach then yes, I think it probably best to roll your own.

Perhaps fatal warnings?

$ perl -ne 'print$ARGV,": ",$_' file1 file2 file3
file1: Hello
Can't open file2: No such file or directory at -e line 1, <> line 1.
file3: World

$ perl -Mwarnings=FATAL,inplace -ne 'print$ARGV,": ",$_' file1 file2 f
+ile3
file1: Hello
Can't open file2: No such file or directory at -e line 1, <> line 1.
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If you want the errors to be kept till after the loop, maybe try a $SIG{"__WARN__"} handler

d:\temp>perl -e"$SIG{"__WARN__"} = sub {push @err,$_[0]; print STDERR 
+$_[0]}; while(<>) {1};print qq<---\n @err>" file1 file2 file3
Can't open file1: No such file or directory at -e line 1.
Can't open file2: No such file or directory at -e line 1.
Can't open file3: No such file or directory at -e line 1.
---
 Can't open file1: No such file or directory at -e line 1.
 Can't open file2: No such file or directory at -e line 1.
 Can't open file3: No such file or directory at -e line 1.

d:\temp>
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This is perl 5, version 32, subversion 1 (v5.32.1) built for MSWin32-x64-multi-thread

you can also only record the filename by pushing $ARGV instead and control what the user sees.

furthermore you can scope the handler with local tightly around the relevant code

{ local $SIG{"__WARN__"}= sub {...}; while(<>){...} }

I could push $ARGV onto an array using eof and compare the result to @ARGV after the loop.

You can't do that because perl removes $ARGV from @ARGV for each file it processes so after the loop @ARGV is empty.

> I could push $ARGV onto an array using eof and compare the result to @ARGV after the loop

please note that the continue will not be executed for empty files, although they exist.

LanX@MDXN MINGW64 /
$ ls -l file*
-rw-r--r-- 1 LanX 1049089 0 Jan 21 23:12 file1
-rw-r--r-- 1 LanX 1049089 2 Jan 21 23:13 file3

LanX@MDXN MINGW64 /
$ perl -e'while(<>){1;}continue{print "---$ARGV\n" if eof;}; print $!'
+ file1 file2 file3
Can't open file2: No such file or directory at -e line 1.
---file3
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Use a file glob:

$ touch file1 file3
$ perl -e'while(<>){1;}continue{print "$ARGV\n" if eof;}; print $!' fi
+le1 file2 file3
Can't open file2: No such file or directory at -e line 1.
$ perl -e'while(<>){1;}continue{print "$ARGV\n" if eof;}; print $!' fi
+le[123]
$
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Is it directed to one-liners only or is it just a demo?

you can always check with -e if the file exists and generate a meaningful message

checking $! "after the loop" seems wrong, because they are reset.

Maybe push them all onto your own @errors_stack which you print afterwards?

edit

> there doesn't seem to be a way for the program to detect that a file wasn't processed

IIRC: $. should tell you the current line number.

You could check if it's false at eof