... %i is not remembered from one iteration to the other.

On the contrary,  %i is remembered perhaps too well, being printed entirely for every line (iteration) in the file. Consider the following modification (my file  words2.txt has the same data as given in the OP):

>perl -nle
"/(\w*)/; $i{$1}++; print qq{$. $_} for keys %i;" words2.txt
1 foo
2 bar
2 foo
3 bar
3 baz
3 foo
4 bar
4 baz
4 qux
4 foo
5 bar
5 baz
5 qux
5 foo
6 bar
6 baz
6 qux
6 foo
7 bar
7 baz
7 qux
7 foo
8 bar
8 baz
8 qux
8 foo
9 bar
9 baz
9 qux
9 foo
[download]

Maybe try something like:

>perl -wMstrict -lne
"/(\w+)/;  $::i{$1}++;
 END { print qq{$_ $::i{$_}} for keys %::i }
"  words2.txt
bar 2
baz 2
qux 2
foo 3
[download]

Update: removed extraneous  BEGIN { ... } block from second example.