I have a 1906 Browning 6.35mm (built in 1905) and it is very close to this one. I would guess this is a very close copy.
As friend eos has said, above, Spanish proofs.
World War One caught everyone a bit (or more) unprepared.... and NOBODY was prepared for anything as long and as horrid, as what actually occurred. Firearms manufacturers everywhere were swamped with contracts and orders, many for small 'hideout' guns such as this one. Spain produced immense numbers of all types with quality ranging from the sublime to the execrable, and they all sold. The French 1916 RUBY 7.65mm pistol, for example, was made by some FIFTY different contractors; I have one here with BRITISH proofs, proudly marked as made by AZANZA Y ARRIZABALAGA, which does have a very Basque-like ring to it.
Spain also produced large-bore revolvers on contract for the British Army, these including the so-called "Old Model" .455, a 6-shot copy of a very early top-break Smith & Wesson.
Another which turns up from time to time is the revolver of the type ARIZMENDI: a copy of a Nagant revolver, 7 shots, in calibres 8mm French AND 7.62mm Nagant. They were built WITHOUT the gas-seal feature, frame and barrel a single part, chiselled from a single block of steel. These are a very solid and workmanlike revolver, but some idiot has Prohibbed them.
BTW, my 1906 Browning is factory-equipped with a LANYARD STUD, so anything is possible.
Your Museum is most fortunate to have one of these very-much-ignored little pieces of History.