The Evacuation of Dunkirk was essentially a defeat (and as close to a rout as it got) for the British Expeditionary Force that had its image turned around due to good PR on the part of the performance of the Royal Navy, Merchant Navy and civilian volunteers who helped shuttle retreating soldiers from Dunkirk, as well as Churchill's famous We shall fight on the beaches speech delivered to the House of Commons on June 4, 1940.
Now despite the fact the BEF was completely surrounded and on the verge of annihilation at Dunkirk, German forces stopped short of annihilating them as they awaited evacuation from the beach. Now the common myth goes that Hitler ordered German Commanders to hold their positions around the BEF rather than finishing them off, though most contemporary historians tend to agree that it was more likely German commanders felt constrained by their supply lines as they had rapidly advanced through France and the Low Countries faster than their supply lines could keep up as well as German forces wishing consolidating their forces before making a final push, giving the BEF time to evacuate as many troops as they could from Dunkirk before the German offensive resumed. Had the Wehrmacht possessed adequate supply lines for the majority of their forces as they reached Calais, (and this veers into what-if territory) it's very likely the entirety if not the majority of the BEF would have been annihilated or captured by German forces. What this would have resulted in is hard to say, though I'd like to think Britain would have still been able to keep herself in the war without concern, as the Germans were not anywhere near well equipped enough to carry out a successful cross-channel invasion. Knowing this, it's clear that the BEF was essentially defeated at Dunkirk, but was sparred the final crushing blow. Though it should be noted, a still considerable number of BEF forces were captured or killed before they were able to be evacuated from Dunkirk, so while the evacuation was seen as a miracle and success, it was not without it's share of setbacks.
The eventual evacuation of Dunkirk of well over 300,000 BEF soliders and personal by the Royal Navy and Merchant Navy as well as civilian volunteers greatly helped turned this defeat into the perfect example of what good PR can do to turn morale or opinion around. Churchill's speech delivered to the House of Commons on the last day of the Evacuation further solidified the idea for the British that the Evacuation of Dunkirk was not so much a defeat as it was a miracle and an example of the British spirit to keep the fight going even when facing overwhelming odds.
Sources:
Inferno: The World at War: 1939-1945by Max Hastings
Depends on how you define triumph.
On one level it can be seen as a triumph in that so many troops were successfully evacuated and the little ships can be seen as a great propaganda scoop.
However on the other hand it can be seen as a complete disaster. For a start most of the British equipment was dumped ranging from tanks to even small arms leaving Britain venerable for quiet a while. There is also the perceived fact that the British concentrated on British soldiers first meaning that most of their allies were left behind.
There is also the fact that the evacuation showed up the limits of the royal navy to evacuation people from a major port (admittedly under air attack) to the point that civilian vessels had to be drafted to assist. Taken out of context such an act would be seen today as military incompetence.
So while at the time it was seen as a triumph it probably was in reality a defeat for Britain.
I think one of the problems in this question is with assigning simplistic words to complex situations. It was a bit of both really. The BEF never had the manpower to resist Germany alone, it was required to fight alongside the French. Once the French collapse started and the Germans had cut the BEF off from the rest of France it was 400,000 or so British troops vs twice that number of German (and with more readily available) and they were in a defensively poor position. Evacuation was inevitable and not specifically a result of any failure of the British Army.
Dunkirk wasnt a traditional X vs Y battle, but an inevitable siege and retreat one, and was of that nature right from the start. Thus, from this point of view it was somewhat of a triumph, as the vast majority of men trapped in the pocket managed to escape. The focus of the Battle of Dunkirk, the victory conditions as it were, wasnt the destruction of the Germans and the reconquest of northern France, but the safe recovery of the valuable troops trapped within. This was largely achieved and that is why this triumph/defeat thing remains an issue. On a broad scale of course it was a defeat, in that it was one of the parts of the Battle of France, which was lost. But in and of itself it was a triumph. A defeat would have to have been the capture or killing of the majority of the men trapped, which didnt occur. Naturally there is no small amount of PR spin around the event, but I still think looking back on it objectively it was a triumphant and successful retreat.
Sources: The Second World War - Sir Winston Churchil