Forget Lady Gaga and her meat dress; no one has ever grasped the art of eccentric showmanship like Sarah Bernhardt, the foremost actress of her age, who liked to accessorise with a dead bat, never went on tour without her own coffin and once travelled round America accompanied by an alligator called Ali-Gaga (who died, sadly enough, as a consequence of consuming too much milk and champagne). Her performances were legendary and her affairs lurid, matched only by a prodigious ability to convert both into hard cash.

In fact, it's a pity she didn't have a small country to run. She might have made a better fist of it than Napoleon III (a one-time lover) or Edward, Prince of Wales (with whom she also enjoyed a brief dalliance). An example of her formidable tenacity occurred during the siege of Paris in 1870, when she co-opted her theatre, the Odéon, and transformed it into a military hospital, filling the dressing rooms, auditorium and stage with cots of injured and dying men and bullying her coterie of lovers into supplying provisions, including, allegedly, the overcoat off one unfortunate's back. Decades later, during the first world war, she elected to have a leg amputated rather than suffer the pain of an old knee injury before launching herself on a gruelling tour of the front, where she entertained the troops in mess tents and ruined barns while hopping about, in her own words, "like a guinea hen".

So much for showmanship. As to her acting, opinion was divided. Victor Hugo (the old roué was another occasional lover) and Lytton Strachey adored her, the latter writing: "She could contrive thrill after thrill, she could seize and tear the nerves of her audience, she could touch, she could terrify, to the top of her astonishing bent." Proust immortalised her as the actress Berma in À la recherche du temps perdu, while George Bernard Shaw despised her, describing her acting as "childishly egotistical" and observing "she does not enter into the leading character: she substitutes herself for it", though he later admitted his enmity might have been because she "was exactly like my aunt Georgina".

Robert Gottlieb is a firmly even-handed biographer and his engagingly zippy account focuses particularly on exposing the cracks in the contradictory stories that Bernhardt and her hagiographers assembled about her life. His modishly open-ended approach reveals Sarah as a protean figure, a skinny self-mythologiser who knew a good story when she saw one and wasn't inclined to let the truth about her origins get in the way of her determination to be the best  and richest  actress alive.

Gottlieb's undercuts are often very entertaining. In her fabulously embroidered memoir My Double Life, Bernhardt tells a devastating tale of saving the widowed Mrs Lincoln from certain death by preventing her from plunging down a flight of stairs, a story that Gottlieb suggests has its origins in a chance encounter on a New York street, during which the two parties were not so much as introduced. The matter of Sarah's own genesis is subject to an equally artful unpicking. She was, it transpires, the illegitimate daughter of a pretty Jewish courtesan who didn't much like, let alone love her, and an absent father whose identity has remained mysterious. She spent her early years with a Breton foster family, went on to a Catholic convent, apparently at the insistence of the absent père, and from there was nudged into stage school by the Duc de Morny, one of her mother's lovers, where she was accepted on account of her exquisite silvery voice and strange, consumptive looks and despite an evident inability to obey rules.

Though she succeeded in freeing herself from the financial anxiety of her mother's way of life, in many ways Sarah didn't stray far from the family blueprint. She also had an illegitimate child at a young age, her beloved Maurice (it's to her credit that she never attempted to conceal this fact). What's more, she relied on a phalanx of well-born benefactors to keep her afloat financially. As Gottlieb tartly observes: "She was leading the life of her mother and her aunt when she wasn't queening it over the French theatre." In later life, Bernhardt corrected this tendency, leaving the safe harbour of the Comédie-Française, France's pre-eminent theatre, in favour of conducting her own whistlestop tours around the globe, performing Phèdre, Hamlet (the title role, naturally, which she made more virile than was then customary) and Tosca to weeping, rapturous crowds. From one such trip, she earned a whopping 2,667,600 francs, partly from ticket sales and partly from the ad campaigns  Pears soap, Columbia bicycles  to which she lent her name.

It's impossible to know now how good Bernhardt really was. The surviving film footage doesn't capture her feverish vitality or her gift for grabbing an audience; her Hamlet, available on YouTube, looks perilously close to hammy. To which she no doubt would have responded with a queenly "quand même?"  so what?  the motto with which she shrugged off all attempts to humiliate or thwart her.

This is a sterling biography, equal to its subject, but it ought to be followed up at once with Bernhardt's own memoirs  and perhaps a jar of smelling salts.