What is the oldest sports trophy?

It is the America's Cup.
http://www.googlism.com/when_is/a/america's_cup/
America's Cup is the oldest trophy in international sport.
America's Cup is the oldest trophy continuously awarded in sport.

The America's Cup is the most famous and most prestigious regatta in the sport of sailing, and the oldest active trophy in international sport, predating the FA Cup by two decades and the Modern Olympics by 45 years. The cup, originally offered as the Royal Yacht Squadron cup, is now named after the first yacht to win the trophy, the schooner America.

The Cup itself is an ornate silver-plated Britannia metal bottomless ewer, crafted in 1848 by Garrard & Co. The regatta's origins date back to August 22, 1851 when the 30.86 m schooner-yacht America, owned by a syndicate that represented the New York Yacht Club, raced 15 yachts representing the Royal Yacht Squadron around the Isle of Wight. America won by 20 minutes.

https://en.wikipedia.org/wiki/America's_Cup

The Stanley Cup, notably the oldest trophy competed for by professional athletes in North America, was donated in 1892 by Sir Frederick Arthur Stanley.

The Silver Trophy Ball of the Grand Base Ball Match of 1861 is historically regarded as the first baseball trophy ever vied for between two teams. It is the earliest known prize ever, that two baseball teams had ever competed for. There are accounts of other trophies that were given to teams for winning either a contest or event but they were given as afterthoughts and not something that teams played and competed for. None of these awards are still known to exist.

Football's - The first FA Cup competition in season 1871-72 had fifteen entries. (This season more than 600 took part.)

The Ashes 1882 is one of the oldest trophies still being fought for. Cricket games of 5 days duration played between England & Australia. Every 2 years England & Australia fight out a 5 or 6 match series, alternating playing in England or Australia.
https://en.wikipedia.org/wiki/The_Ashes