<p>We investigate in this paper the so-called pointed Shafarevich problem for families of primitive symplectic varieties. More precisely, for any fixed pointed curve (<i>B</i>,&#xa0;0) and any fixed primitive symplectic variety <i>X</i>, among all locally trivial families of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Q}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>-factorial and terminal primitive symplectic varieties over <i>B</i> whose fiber over 0 is isomorphic to <i>X</i>, we show that there are only finitely many isomorphism classes of generic fibers. Moreover, assuming semi-ampleness of isotropic nef divisors, which holds true for all hyper-Kähler manifolds of known deformation types, we show that there are only finitely many such <i>projective</i> families up to isomorphism. These results are optimal, since we can construct infinitely many pairwise non-isomorphic (not necessarily projective) families of smooth hyper-Kähler varieties over some pointed curve (<i>B</i>,&#xa0;0), such that they are all isomorphic over the punctured curve <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(B\backslash \{0\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="true">\</mo> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and have isomorphic fibers over the base point 0.</p>