Sending data through broken pipe

When I connect a socket to a server socket, and the server socket at a given time shuts down, I get a BrokenPipeError on the client side. But not the next time I try to send something, but the time after that.

Here a SSCCE:

Server:

#! /usr/bin/python3

import socket

s = socket.socket (socket.AF_INET, socket.SOCK_STREAM)
s.setsockopt (socket.SOL_SOCKET, socket.SO_REUSEADDR, 1)
s.bind ( ('', 10100) )
s.listen (1)
print ('Waiting on client')
client, _ = s.accept ()
print ('Accepted')
data = b''
done = False
while not done:
    data += client.recv (4096)
    msgs = data.split (b'r')
    for msg in msgs [:-1]:
        print ('received {}'.format (msg) )
        done = msg == b'exit'
    data = msgs [-1]
s.close ()
print ('Server down')

Client:

#! /usr/bin/python3

import socket

s = socket.socket (socket.AF_INET, socket.SOCK_STREAM)
print ('Connecting')
s.connect ( ('localhost', 10100) )
print ('Connected')
for msg in [b'ping', b'pang', b'exit', b'ping', b'pang']:
    print ('Sending {}'.format (msg) )
    sent = s.send (msg + b'r')
    print ('Sent {}. {} bytes transmitted'.format (msg, sent) )
    input ('>> ')

I start up the server, then the client and hit enter to step through the messages.

The server output is:

Waiting on client
Accepted
received b'ping'
received b'pang'
received b'exit'
Server down

The client output is:

Connecting
Connected
Sending b'ping'
Sent b'ping'. 5 bytes transmitted
>> 
Sending b'pang'
Sent b'pang'. 5 bytes transmitted
>> 
Sending b'exit'
Sent b'exit'. 5 bytes transmitted
>> 
Sending b'ping'
Sent b'ping'. 5 bytes transmitted
>> 
Sending b'pang'
Traceback (most recent call last):
  File "./client.py", line 10, in <module>
    sent = s.send (msg + b'r')
BrokenPipeError: [Errno 32] Broken pipe

Why do I get the BrokenPipeError after the last pang and not after the ping?

Why does send return 5 when sending the ping after the exit?

Why is the pipe not broken immediately after the server is down?

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EDIT: After having sent exit, I don’t hit enter on the client console unless the server console has already printed Server down.