As some of us have been saying for several hours, it looks as if overnight something happened to change the Incoming Mail Server address, which as I recall was "imap.mail.me.com" to "p99-imap.mail.me.com."
Those of us having problems with the Apple Mail client cannot change this Incoming Mail Server designation to anything else except the latter of the two quoted mail-server names in the previous paragraph because, as has been said repeatedly, the box is greyed-out and uneditable.
My Outgoing Mail Server is set to "<my mac.com address> (iCloud)" as I believe it always has been since I started using iCloud, and "Use only this server" is unchecked, as was suggested by someone in some Mac support forum earlier today. As others have noted, I too can send outgoing mail, which appears in my iCloud Mail when accessed through the Web.
As also noted previously, changing the IMAP Path Prefix box doesn't help. The problem, to reiterate, seems to be caused because those of us unable to use the Apple Mail client cannot change the Incoming Mail Server designation to anything other than "p99-imap.mail.me.com"--a change that appears to have been made somehow at Apple's mail-server end, not at any of our individual accounts. And again, we simply can't edit this box because we're not allowed to do so.
For anyone out there who might have a solution, here's a complete list of information under Account Information < Accounts < Preferences for my Apple Mail account:
Account Type: iCloud IMAP
Description: <my mac.com address>
Email Address: <my mac.com address>, <my me.com address>
Full Name: <my full name>
Incoming Mail Server: p99-imap.mail.me.com
User Name: <my username>
Password: <my password>; NOTE: I checked that this works, and it does. I retyped it as well. Still no luck.
Outgoing Mail Server (SMTP): <my mac.com address> (iCloud)
Use Only This Server: <unchecked>
TLS Certificate: None
And a complete list of information under Advanced < Accounts < Preferences:
Enable this account: <checked>
Include when automatically checking for new messages: <checked>
Compact mailboxes automatically: <checked but greyed out>
Keep copies of messages for offline viewing: <All messages and their attachments>
IMAP Path Prefix: <blank>
Port: 993
Use SSL: <checked>
Authentication: <Password>
Use IDLE command if the server supports it: <checked>