In the sense of the blade-wielding master of espionage who could plow through a dozen guard single-handedly? No. That, unfortunately, is something that only exists in the realm of legends and entertainment media.

The word "assassin," as you might already know, entered into many European languages via the Crusades as a bastardization of the Arabic حشاشين‎ (hashishiyyin), referring to a sect of Nizari Isma’ilis in present-day northern Iran who orchestrated a series of political assassinations in the 1100s. The name is supposed to be a derogatory term meaning “hashish-consumers,” deriving from the belief that Nizari fida’is (devotees) were worked into a frenzy through the consumption of the drug. However, most scholars recognize that there is no proof of this. The story comes from Marco Polo’s writings, specifically a section entitled “Concerning the Old Man of the Mountain” (Book 1, Chapter 23), where he claimed he heard from some natives that an old man in Mulehet drugged his followers and secluded them in a garden to trick them into believing they could enter Paradise:

He kept at his Court a number of the youths of the country, from 12 to 20 years of age, such as had a taste for soldiering, and to these he used to tell tales about Paradise, just as Mahommet had been wont to do, and they believed in him just as the Saracens believe in Mahommet. Then he would introduce them into his garden, some four, or six, or ten at a time, having first made them drink a certain potion which cast them into a deep sleep, and then causing them to be lifted and carried in. So when they awoke, they found themselves in the Garden.

There haven’t been any other sources to substantiate this. Historians even disagree on the etymology of the word hashishiyyin. Farhad Daftary, for example, claims that the word, though used pejoratively, had no true connection to the drug hashish. It was simply a derogatory word used by Nizaris for other Muslims that the Crusaders picked up in the Levant, and the assassin myths sprang up along with a number of rumors regarding secret practices of the Nizari Isma’ilis (10-11). The word picked up the connotation of professional killer around the mid-14th century. Dante, for example, spoke of "Le perfido assassin" in the Inferno. “Assassin” up to that point was exclusively associated with Muslims, and even after it was often used in the plural by Orientalists.

So to return to the original question - whether there were types of trained killers, mythic or otherwise, in the medieval European context - it doesn’t seem likely. The most effective assassins are those who can get close to their targets without arousing suspicion and kill him/her with as little risk to themselves as possible. In other words, you didn’t need to be trained as sword-wielding acrobat with a devil-may-care attitude to be an effective assassin.

Instead of death by cold steel, the most common method of assassination in medieval Europe by far was poisoning. Poisons were cheap and relatively easy to obtain, and poisoning is a clever way to conceal assassination as an illness (most poisons wouldn’t cause you to drop dead on the spot - at least not if the perpetrator was doing it right). There were many herbal poisons readily available (e.g. Nightshade, Water Hemlock, Laburnum, or Black Hellbore), but there were also more deadly chemicals such as arsenic powder. Deliberate poisoning of food was common enough in Europe that it was often widely believed to be the cause of many royal deaths, whether or not it was true. When King John of England died in 1216 of dysentery, for example, numerous rumors began almost immediately that he had been poisoned by a monk. There were many visual images of this poisoning that circulated for years to come, both in prints and in illuminated manuscripts. There were so many rumors about Lucrezia Borgia and her brother Cesare poisoning their enemies that they were said to own rings with secret compartments used to casually slip poison unnoticed into glasses. There isn’t much evidence to link them directly to this, but it shows the extent to which the idea of assassination was linked to poison. And yes, some affluent individuals did employ food-testers as a deterrent against this kind of risk.

Anyone who had extensive knowledge of poisons, then, might be considered a potential assassin (or at least able to aid a potential one). With the printing press, pharmacology manuals detailing poisons became more widely available; Magister Santes de Ardoynis’s The Book of Venoms (1424) was probably the most popular. However, before then most major cities contained apothecary guilds comprising a large number of tradesmen knowledgeable in poisons. Apothecaries, of course, were not simply poison-sellers, but rather served as both pharmacist and general medical practitioner in the Middle Ages. In many ways, apothecaries were a vital resource for medicine in medieval Europe. So this raises a question: why did apothecaries sell poisons at all? Well, in their view they didn’t. It was a long-standing belief in many parts of Europe that certain plant extracts which were poisonous in large doses were beneficial to one’s health in smaller doses. For example, Henbane - the poison Claudius uses to murder the former king in Shakespeare’s Hamlet - was sometimes recommended in small doses as a sleeping agent or as a sedative for hysteria. The same items could be used as a form of pest control. Mix Aconite with animal fat and/or honey and you have an effective way to kill a wolf or a fox threatening your livestock.

All of these concoctions, then, had designated uses other than murdering human beings and were sold as medical remedies. But they could be used to commit murder in large doses. The trick was knowing what the right dose was to induce death without making it obvious that the victim had been poisoned.

EDIT: fixed a broken link and a typo.

I think I can probably answer the question by answering a slightly different question. It hedges on vioating the 20 year rule, but this is really just illustrative.

Suppose you were to ask the question, "Do professional assassins exist in the present day?"

The answer to this question depends on how you mean "assassin." In the terms that you seem to be asking, the answer is almost certainly no. There are certainly soldiers trained to a high level of skill etc. and while I have little doubt there are some people who kill for criminal organizations, these are by and large people that do not keep records of the sort that a historian might ever find, except for those that get caught. A group of "professional assassins" that is known could not openly survive for long. Criminal "assassins" are usually far closer to thugs than "professionals" in any case. Usually it's find an excuse to get close, then open up with a pistol, going down in the process.

Rather, what you find is that when "assassinations" happen in the modern world, it's usually a lone gunman or bomber who is not particularly "professional," and dies in the attempt. Sometimes it's a whole team of operatives, but then they're distinctly military or paramilitary rather than some covert group. (see. e.g. Benazir Bhutto's assassination).

Even though the historical context changes dramatically, I think you can compare modern times to historical times in this regard. Most assassinations in the historical record were not performed by someone you would describe as a "professional."

For proof, take a look at european assassinations

Louis, Duke of Orleans 1407 - subterfuge lures the duke out into the open where 15 masked thugs stab and beat him.

Henry III, King of France, 1575 - A domnican friar, Jaque Clement, gains admittance to the king's presence with documents, saying he has a message for the king. He stabs the king once and is killed on the spot by the guards. the king later dies of the wound.

Henry IV, King of France, 1610 A catholic fanatic Francois Revellec waits until the king's carriage is stopped by a traffic blockage (possibly arranged by compatriots) climbs into the carriage and stabs the king. He is immidiately arrested, tried and executed.

Phillip v Hohenstauffen "King of the Germans" and Prince of Swabia 1208 - - the king granted an audience to another noble, Otto VIII v. Wittlesbach, who drew his sword and stabbed the king in the neck. (Apparently over a dispute caused by a broken wedding engagement). Wittlesbach fled and then was tracked down and killed.

Guliano de'Medici, Duke of Florence, 1478 was killed in the "pazzi conspiracy" - a plot to remove the Medici family as the rulers of Milan, Bernardo Bandi and Francesco de' Pazzi, themselves italian nobility, wait until Guliano d'Medici is attending church, attack him and stab him 19 times.

William I of Orange, 1584 - After William gained independance for the Netherlands, Philllip II of spain declared him an outlaw and promised a bounty of 25,000 Crowns. Balthasar Gerard, a catholic frenchman, made an appointment with William, gained an audience, shot William in the chest at close range with a wheel lock pistol, then fled.

While that's not exhaustive, I think that gives a pretty good selection of medieval era assassinations. By and large they're not all that different than you might expect an assassination to be today. It's far more likely to be some crazy with a knife or a gun, dying in the attempt, than some highly trained operative who kills and vanishes.

Edit: fixed various spelling and grammatical errors. Thanks to /u/Sparadise for the correction.

If I may ask a question in this thread relevant to the main question -

Did any of the holy orders have people trained to fulfill the roles of assassins/spys during the medieval period?

I have a follow-up question of sorts: I know a little (very little; dangerously little) about ninja in medieval Japan and their counterparts in China -- they were basically groups for hire, "muscle" if you will, but trained for the job. That's as far as my knowledge goes, and it could be totally wrong.

The question (other than hoping for any clarification on the above), is, were there any comparable groups in medieval Europe? Mercenary groups that trained independently of militaries for the purpose of hiring services out. I'm hoping this question makes any sense at all.