Non-deterministic programming languages
One example of a non-deterministic language is Occam, based on CSP theory. The combination of the PAR and ALT constructs can give rise to non-deterministic behaviour in multiprocessor systems, implementing fine grain parallel programs.
When using soft channels, i.e. channels between processes on the same processor, the implementation of ALT will make the behaviour close to deterministic†, but as soon as you start using hard channels (physical off-processor communication links) any illusion of determinism vanishes. Different remote processors are not expected to be synchronised in any way and they may not even have the same core or clock speed.
†The ALT construct is often implemented with a PRI ALT, so you have to explicitly code in fairness if you need it to be fair.
Non-determinism is seen as a disadvantage when it comes to reasoning about and proving programs correct, but in many ways once you've accepted it, you are freed from many of the constraints that determinism forces on your reasoning.
As long as the sequencing of communication doesn't lead to deadlock, which can be done by applying CSP techniques, then the precise order in which things are done should matter much less than whether you get the results that you want in time.
It was arguably this lack of determinism which was a major factor in preventing the adoption of Occam and Transputer systems in military projects, dominated by Ada at the time, where knowing precisely what a CPU was doing at every clock cycle was considered essential to proving a system correct. Without this constraint, Occam and the Transputer systems it ran on (the only CPUs at the time with a formally proven IEEE floating point implementation) would have been a perfect fit for hard real-time military systems needing high levels of processing functionality in a small space.
The Wikipedia article points to Amb which is a Scheme-derivative with capacities for non-deterministic programming.
As far as I understand, the main reason why programming languages do not do that is because running a non-deterministic program on a deterministic machine (as are all existing computers) is inherently expensive. Basically, a non-deterministic Turing machine can solve complex problems in polynomial time, for which no polynomial algorithm for a deterministic Turing machine is known. In other words, non-deterministic programming fails to capture the essence of algorithmics in the context of existing computers.
The same problem impacts Prolog. Any efficient, or at least not-awfully-inefficient Prolog application must use the "cut" operator to avoid exploring an exponential number of paths. That operator works only as long as the programmer has a good mental view of how the Prolog interpreter will explore the possible paths, in a deterministic and very procedural way. Things which are very procedural do not mix well with functional programming, since the latter is mostly an effort of not thinking procedurally at all.
As a side note, in between deterministic and non-deterministic Turing machines, there is the "quantum computing" model. A quantum computer, assuming that one exists, does not do everything that a non-deterministic Turing machine can do, but it can do more than a deterministic Turing machine. There are people who are currently designing programming languages for the quantum computer (assuming that a quantum computer will ultimately be built). Some of those new languages are functional. You may find a host of useful links on this Wikipedia page. Apparently, designing a quantum programming language, functional or not, and using it, is not easy and certainly not "simple".
Prolog is actually deterministic—the order of evaluation is prescribed, and order matters.
Why isn't nondeterminism more popular?
Nondeterminism is unpopular because it makes it harder to reason about the outcomes of your programs, and truly nondeterministic executions (as opposed to semantics) are hard to implement.
The only nondeterministic languages I'm aware of are
Dijkstra's calculus of guarded commands, which he wanted never to be implemented
Concurrent ML, in which communications may be synchronized nondeterministically
Gerard Holzmann's Promela language, which is the language of the model checker SPIN
SPIN does actually use the nondeterminism and explores the entire state space when it can.
And of course any multithreaded language behaves nondeterministically if the threads are not synchronized, but that's exactly the sort of thing that's difficult to reason about—and why it's so hard to implement efficient, correct lock-free data structures.
Incidentally, if you are looking to achieve parallelism, you can achieve the same thing by a simple map function in a pure functional language like Haskell. There's a reason Google MapReduce is based on functional languages.