No, there is a 5V to 3.3V regulator on the module. The Martyn Currey tutorials linked above really are the best.
Do you realize that this loop floods the "air waves" with a nearly continuous stream of 1s and 0s? That flood of data might cause the disconnection.
// Reading the button
buttonState = digitalRead(button);
if (buttonState == HIGH)
{
Serial.write('1'); // Sends '1' to the master to turn on LED
}
else
{
Serial.write('0');
}
delay(10);
It would make much more sense to send the switch state at longer intervals, or better, when the switch state changes.