Abstract A mapping T : A ∪ B → A ∪ B {T\colon A\cup B\to A\cup B} such that T ⁢ ( A ) ⊆ A {T(A)\subseteq A} and T ⁢ ( B ) ⊆ B {T(B)\subseteq B} is called a noncyclic mapping, where A and B are two nonempty subsets of a Banach space X. A best proximity pair ( p , q ) ∈ A × B {(p,q)\in A\times B} for such a mapping T is a point such that p = T ⁢ p , q = T ⁢ q {p=Tp,q=Tq} and ∥ p - q ∥ = dist ⁡ ( A , B ) {\|p-q\|=\operatorname{dist}(A,B)} . In the current paper, we establish some existence results of best proximity pairs in strictly convex Banach spaces. The presented theorems improve and extend some recent results in the literature. We also obtain a generalized version of Markov–Kakutani’s theorem for best proximity pairs in a strictly convex Banach space setting.