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We prove an interpolation theorem for slice-regular quaternionic functions. We define very tame sets in    $${\mathbb {H}}^2$$  H 2 to be the sets which can be mapped by compositions of automorphisms with volume 1 to the set    $${{\mathcal {T}}}=\lbrace (2n-1,0), n \in {\mathbb {N}}\rbrace \cup \lbrace (2n + {\mathbb {S}},0), n \in {\mathbb {N}}\rbrace .$$  T = { ( 2 n - 1 , 0 ) , n ∈ N } ∪ { ( 2 n + S , 0 ) , n ∈ N } . We then show that any zero set of an entire slice-regular function of one variable embedded in    $${\mathbb {H}}\times \lbrace 0 \rbrace \subset {\mathbb {H}}^2$$  H × { 0 } ⊂ H 2 is very tame in    $${\mathbb {H}}^2.$$  H 2 . A notion of slice Fatou–Bieberbach domain in    $${\mathbb {H}}^2$$  H 2 is introduced and, finally, a slice Fatou–Bieberbach domain in    $${\mathbb {H}}^2$$  H 2 avoiding    $${{\mathcal {T}}}$$  T is constructed in the last section.

An interpolation theorem for slice-regular functions with application to very tame sets and slice Fatou–Bieberbach domains in $${\mathbb {H}}^2$$