  This work is devoted to the advanced study of Roper--Suffridge type extension
operators. For a given non-normalized spirallike function (with respect to an
interior or boundary point) on the open unit disk of the complex plane, we
construct perturbed extension operators in a certain class of Banach spaces and
prove that these operators preserve the spirallikeness property. In addition,
we present an extension operator for semigroup generators.
  We use a new geometric approach based on the connection between spirallike
mappings and one-parameter continuous semigroups. It turns out that the new
one-dimensional covering results established below are crucial for our
investigation.
