Sure. For example, a communal village often has a village chief that was selected by the majority of the inhabitants (by definition, a democratic election). In this case, the village chief is the "head of state" (state being the village) and represents the village (e.g. to a feudal lord such as a baron or daimyo).
Everyone seems to make up their own definition of democracy (and other governments) but since it's well known, I'll use Machiavelli's definitions. Machiavelli defines any state as a republic, but differentiates between rule of the one (monarchy), rule of the few (aristocracy), and rule of the many (democracy). In all three cases, he does emphasize that you can have both an unelected head and an elected head, so the method of choosing your leader does not make one a democracy or not.
This of course is obvious. Prior to World War II, nationalist parties were legitimately voted in and transitioned to a monarchal form. You can see easy examples with with Stalin, Mussolini, and yes, Hitler.
Machiavelli often uses the Roman Republic as an example when speaking about republics with elected representatives. The Roman Republic had a diarchy: that is to say that every year they elected two people (consuls) to lead the nation. They, in this case, being the senatorial class. However, if you argue that excluding some segment of the population from a vote makes it not a democracy, then no nation was ever a democracy. Non-citizens, non-humans, infants, toddlers, other children, slaves, etc. could not (and can not) vote. The consuls represented the people of Rome (take that any way you wish).
Another example, coincidentally contemporaneous with Machiavelli, is the Holy Roman Empire (no relation to the Roman Empire). It was more of a confederacy, a collection of sovereign states, each with their own monarchs. The aristocrats would come together every few years to select the Holy Roman Emperor by majority vote (by definition, a democratic election). Sure, the emperor is similarly elected by his peers, but in no democracy is everyone eligible for office. For example, no country allows a 4 year old to be president (however, monarchies do allow 4 year olds to be sovereign).
Another example is a diocese, or ecclesiastical district. A bishopric is, as the name implies, headed by a bishop. In Eastern Catholic churches, bishops (who are monarchs in a diocese) are elected. Similarly for Catholicism in general, the papacy is also an elected position (one again, bishops choosing from among their peers). Bishops (which include the pope) represent the catholic institution. Since bishops were historically monarchs, this also makes the Catholic church an institution similar to the mechanisms of the (contemporaneous) Holy Roman Empire, where monarchs already in a hierarchy elect a top level monarch amongst themselves each time the previous one dies.
A common theme you may see is that people, such as those with political power, will select a leader from amongst themselves, not including the rest of the population. But you will realize that this is fundamentally identical to the selection a prime minister. Thus, if you accept that a parliamentary system is democratic, then you much also logically accept that these historical systems were also democratic. One can make the parallel that one must amass political power to be a party member in parliament (that is to say, a layman member of the population cannot join parliament) in the same way that one must amass political power in the Roman senate. The differences between the Roman senate, the aristocrats of medieval Europe, and the clergy of Christianity is the path one takes to acquire political power, whether by economic wit, military success, or piety. Similarly, those in modern systems often chose rhetoric as a way to acquire political power.
As a bonus, the French Republic (that is, the country that was in the area of modern France post French Revolution, just after the Directory Government, which followed the Reign of Terror) elected three consuls (a triarchy). While this was bound to fail (following the Roman Empire's example of a tetrarchy with four emperors) because it is inherently unstable, it failed anyway because Napoleon decided to follow the route of Julius Caesar (almost verbatim): eliminating the other consuls (his other political rivals, which were also called consuls), declaring himself consul for life (well, technically Caesar became dictator, a different political office in Rome, for life), and becoming a dictator (although Caesar became a dictator through legitimate means and was appointed). No system, even that which the leaders are elected from the populace, is immune to descent into tyranny. Although Machiavelli proses that monarchy descends to tyranny, aristocracy descends to oligarchy, and democracy descends to anarchy.
