Do all sounds that have a pitch have a fundamental frequency?

All sounds are composed of periodic waves, but if there are too many frequencies and not a single one prevails , you will not hear it as a sound, but as noise . So you are right, to have a pitch you have to have a prevailing frequency .

I'm not aware of any purely physical notion of "pitch". Pitch has something to do with physical frequencies but also how your brain perceives the sound. The overtones accompanying a fundamental frequency, if any, may influence the perceived pitch in a way similar to how a given color (say, a single frequency of light) may appear different depending on what colors are around it. You can also get cases where two pure frequencies played together are perceived as more than two pitches, which I think gets at your question more directly.

Pipe organs might be a particularly interesting instrument to check out for this type of question. For a single pipe, pitch is primarily determined by length, but the organ is usually designed with stops that allow multiple pipes to be played together on the press of single key. There are different schemes for choosing the stops to alter the subjective qualities of the music played, including but not limit to dealing with pitch.

The so-called "missing fundamental" is a more precise example, which is used among other places in organ design to extend the range of pitches that a listener experiences without including longer and longer pipes. When the overtones are present, the mind will fill in the fundamental even though it is not physically produced.

Some more information here: https://en.wikipedia.org/wiki/Pitch_(music)

And here: https://en.wikipedia.org/wiki/Missing_fundamental