<p>The main aim of this paper is to study multidimensional Bohr radii for holomorphic functions defined in complete Reinhardt domains in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> with values in complex Banach spaces. More specifically, for holomorphic functions with values in arbitrary complex Banach spaces, we explore the asymptotic estimates of the classical Bohr radius and arithmetic Bohr radius in the unit ball of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\ell ^n_q\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>ℓ</mi> <mi>q</mi> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((1\le q\le \infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>≤</mo> <mi>q</mi> <mo>≤</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> spaces. Further, we study a mixed version of Bohr radii for vector-valued holomorphic functions and as a consequence we obtain the exact value of mixed arithmetic Bohr radius.</p>