How much evidence is there regarding the naval punishment of keelhauling?

According to this source (wiki) a keelhauling could take place over either the length or the width of a ship. A keelhauling over the length would be fatal, either through drowning, or through lacerations brought by contact with the ship.

A keelhauling across the width (typically about one third of a ship's length) was a "lesser" punishment that might give the victim a fighting chance to survive. For instance, the dimensions of a sloop (a small ship) was something like 39 feet long and 14 feet width. It's much easier to imagine someone surviving over 14 feet than 39 feet.

The practice existed, because there are documented instances, as another poster pointed out. For all that, the lack of records suggests that it was rare. It was administered for only the most egregious offenses, such as mutiny or extreme cowardice. More common forms of execution were hanging, or shooting, or in the case of pirates, forcing the victim to "walk the plank." To estimate the frequency of keelhauling, one can start with the more common forms of execution (not so frequent in themselves) and work "downwards" from there.

I was interested enough in my own question to reach out to a naval historian at Oxford University, Nicholas Rodger, to see if he could shed any light on the subject. His answer was, indeed, illuminating and confirmed my supposition. I'm reproducing it here, largely verbatim, for general interest.

Several 17th-century English writers like Monson (Naval Tracts ed. Oppenheim III,436) and Boteler (Dialogues ed. Perrin pp.11-25) thought keel-hauling had once been practiced in English ships – before their time, but they were vague about the date. I know no authentic evidence for it at all.

It was an official (though rare) punishment in the Dutch navy (Bruijn, Dutch Navy p.140), it is claimed as a custom among pirates, and the French occasionally used la cale, which seems to have been just ducking from the yardarm. William Spavens, who served in Dutch ships, describes ‘this barbarous custom’, apparently from life (Narrative, Folio Soc edn p.110), but he describes it - in line with Falconer - as essentially a prolonged ducking.

My instinct would be to receive stories of barbarous cruelties with extreme scepticism, except relating to the Russian navy or to pirates, when they may well be true.

The earliest evidence for keel-hauling that I'm aware of is actually from ancient Rhodes (~800BC) in the Lex Rhodia. There are also depictions of the practice known from ancient Greece (one is included as the frontispiece of Henry Omerod's Piracy in the Ancient World).

As the Wikipedia page notes, the Dutch Navy introduced the punishment in an ordinance dated 1560, although it may have already been used prior to that date. They didn't actually abolish the practice until 1853. The keel-hauling of the ship's surgeon of Admiral Jan van Nes in the late 1600s was commemorated in a painting by Lieve Verschuier now held at the Rijksmuseum in Amsterdam. Records of punishments meted out during a voyage would be recorded by the individual ship's Captains. As far as I'm aware, no serious analysis has yet been carried out on the surviving records. I believe these are held at the National Archives of the Netherlands, but are not available online (and would - obviously - be written in Dutch).

The Royal Navy certainly did use it as a punishment, but I don't know how widespread the practice was. I've only seen a couple of instances (both were fatal). Again, instances of keel-hauling would be recorded in the individual ship's logs, together with the punishments meted out for other infractions. Many of these are held at the National Archives in Kew, as Captain's logs, Supplementary Logs, or Ships logs. Most of these would need to be viewed in person as they are not yet available online. (It will probably make for part of an interesting PhD research project for an aspiring historian some day).