How to suppress insertion of letter labels after year for references from the same year

Find the file apalike.bst in your TeX distribution (likely in a spot such as .../bibtex/bst/apalike, where ... is a root directory of the TeX distribution), make a copy of it and name the copy, say, my-apalike.bst. Open the new file in an editor of your choice and search for the function FUNCTION {output.year.check} (probably starting around line 120 of the file). In this function, locate the line:

    " (" year * extra.label * ")" *

Remove the substring extra.label *. I.e., the line should now read as follows:

      " (" year * ")" *

Save the file (either to your current working directory or in some directory that's searched by TeX's filename database operations, and update the TeX filename database if necessary) and be sure to use the instruction

\bibliographystyle{my-apalike}

from now on.