Restated in more words, would a conscripted pikeman with a decent set of chainmail but no shield have any hope against an arrow from your basic english longbow?
This depends a lot on the arrow and type of chainmail. the draw of the bow (english longbows do not have a standardized draw) also matters. A broadhead hitting an 8 ring mail wouldn't yield much more than a scratch, a spike bodkin will go right through 4 ring mail and barely notice the armor. Image of different chainmail types below. 4 ring is by far the most common (~97-98%) but examples of 6 and 8 ring do exist. there was even something called doublemail but no one knows what it was, I personally suspect it is either 8 in 1 or another name for kingsmail a variation of 4 ring in which every ring is doubled called 8 in 2. Keep in mind you don't wear chainmail by itself but with padding below it which also matters.
If the archers know there will be armored targets they will be using bodkins which are very good at penetrating chainmail (and armor in general) of course the downside of these arrowheads is they don't cause all that much bleeding meaning they take time to kill someone most of the time.
Broadhead arrows - definitely, it would.
Bodkin arrow - probably yes. It depends on many factors like distance between archer and his target, angle of impact, draw of bow, etc.
But even if an arrows penetrates the mail, it will not kill the soldier wearing it. Because of gambeson under the mail. And gambeson is an amazing thing, it can deflect blows even arrows (if you are lucky), it´s cheap, available and easy to produce.
It´s also possible to wear second gambeson over the mail to increase protection.
If the pikeman was 20 yards farther than the maximum range of the bow he would be safe from that shot since it wouldn't even reach him.
In the US Civil War bullets would slow down as they traveled farther and farther and would eventually be what was called "spent rounds" shortly before falling to the ground. Many soldiers survived being hit by spent rounds without serious injury.
If the same thing happened to arrows then there would be distances where arrows from even the strongest bows couldn't reach, closer distances where unarmored men wouldn't be seriously injured by arrows, closer distances where men in mail were safe but unarmored men would be seriously wounded, closer distances where arrows penetrated mail and inflicted serious wounds but couldn't penetrate plate armor, and perhaps closer distances where arrows might penetrate even plate armor.
And perhaps someone who is more of an expert on medieval archery will be able to tell you what those ranges would be.