Incomparable ω₁-like models of set theory

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We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω₁-like models of set theory. Specifically, under the hypothesis and suitable consistency assumptions, we show that there is a family of 2ω¹ many ω₁-like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω₁-like model of ZFC that does not embed into its own constructible universe; and there can be an ω₁-like model of PA whose structure of hereditarily finite sets is not universal for the ω₁-like models of set theory.