There were two unusual entertainments for the crew and passengers on board the Red Dragon on 5 September 1607, according to a tattered and stained page of the captain’s diary: “We had the Tragedy of Hamlet: and in the afternoon we went altogether ashore, to see if we could shoot an elephant.”

Now on display in a major Shakespeare exhibition at the British Library, if the diary of Captain Keeling is genuine, it is the first record anywhere in the world of an overseas – literally, in this case – performance of a play by the bard.

The audience included a Portuguese translator and some African merchants invited on board in the hope of some trade. The sailors evidently enjoyed the sword fights and soliloquys, as Keeling went on to record a performance of “Kinge Richarde The Seconde” three weeks later. And a few months later it was Hamlet again.

“I invited Captain Hawkins to a fyshe dinner,” he wrote on 3 March 1608, “and had Hamlet acted aboard, which I permit to keep my people from idleness and unlawful games, or sleepe.”

When the battered remains of his diary were discovered in the 19th century, the image it created of wholesome leisure time on a 17th-century merchant ship bound for Java was so startling – given that Hamlet was probably first performed in London only in 1603 – that some suspected it was a forgery. If so, it is in good company in the exhibition which, among records of 400 years of Shakespeare in performance, includes many fabulous literary discoveries with dubious pedigrees.

The playbill and script for the 18th-century London performance of Vortigern are rarer than those for a genuine Shakespeare play. It was forged by an enterprising young man called William Henry Ireland, together with documents explaining how it came to be in his family – gratitude for saving the Bard of Avon from drowning, apparently – and, for good measure, a letter to Anne Hathaway signed “Thynne everre Wm Shakspeare”.

The Tragedy of Vortigern and Rowena survived one night at Drury Lane before it was literally laughed off the stage: contemporary accounts say that at one point the show had to be stopped for 10 minutes to allow the laughter and uproar to die down.

Curator Tanya Kirk said that in deciding to mark the 400th anniversary of Shakespeare’s death, the library first thought of doing something conventionally biographical. “But what new is there to say about the man? There is so little information about him. What we did have an absolute treasure trove of material about what made him famous, performances of his work.”

The exhibition includes posters, scripts, props, costumes, paintings, recordings and films of some of the most famous performances and productions of the plays, including a chance to compare nine versions of Hamlet’s “To be or not to be”, from Herbert Beerbohm Tree in 1906 to David Tennant in 2013, via Laurence Olivier and John Gielgud.

A skull on display was given by the French author Victor Hugo – complete with his own verse written on the bone – to the great tragic actor Sarah Bernhard. She used it as Yorick’s skull when she starred as Hamlet in 1899, aged 55, in London and Stratford-upon-Avon.

Women arrived on the Shakespearean stage with the restoration of Charles II, who licensed them as actors for the first time. The exhibition includes an illustration of Edward Kynaston, one of the last men to earn his living playing Shakespeare’s women. The diarist Samuel Pepys said he was “clearly the prettiest woman in the whole house”.

Kirk particularly likes another famous tragic actor, Sarah Siddons, who, when unhappy with some of her portraits, took up sculpture herself to set the record straight. An oil painting records her as Lady Macbeth, swathed in voluminous draperies. “Did people really sleep swathed like that,” Kirk wondered. “Could she even move properly on stage in all that?”

Two centuries later the uproar was over a much less covered-up Lady Macbeth: Francesca Annis played the sleepwalking scene in Roman Polanski’s 1971 film wearing nothing but her talent.

The critic Kenneth Tynan’s furious letter to the papers is also in the exhibition. He had worked on the script of the film, and insisted that in Shakespeare’s day people slept naked.

“Whether or not Shakespeare wanted Lady Macbeth to wear anything in the sleepwalking scene is a question worth debating, since he had no choice in the matter. He had to prescribe some form of covering because the role was played by a boy. Happily Mr Polanski is working under no such limitation.”

• Shakespeare in Ten Acts is at the British Library in London until 6 September.