Actually Wimbledon does not play as it is not a person.The tennis players only play in the Wimbledon championship in the All England Club which was in Wimbledon district.
Yes, Jan Fallis did play at wimbledon. I know her personally.
Rafael Nadal did not play in the 2009 Wimbledon championships due to a knee injury.
Andy Murray reached the Wimbledon semis in 2009.
Sardar Nihal Singh was the first Indian player, who played the Wimbledon in 1908.
They were allow to play since the origin of the tournament.
ANSWER:Three countries that play tennis are Switzerland, Wimbledon and Australia.