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For an algebraically closed non-archimedean extension $C/\mathbb{Q}_p$, wedefine a Tannakian category of $p$-adic Hodge structures over $C$ that is alocal, $p$-adic analog of the global, archimedean category of$\mathbb{Q}$-Hodge structures in complex geometry. In this setting thefiltrations of classical Hodge theory must be enriched to lattices over acomplete discrete valuation ring, Fontaine's integral de Rham period ring$B^+_\mathrm{dR}$, and a pure $p$-adic Hodge structure is then a$\mathbb{Q}_p$-vector space equipped with a $B^+_\mathrm{dR}$-latticesatisfying a natural condition analogous to the transversality of the complexHodge filtration with its conjugate. We show $p$-adic Hodge structures areequivalent to a full subcategory of basic objects in the category of admissiblepairs, a toy category of cohomological motives over $C$ that is equivalent tothe isogeny category of rigidified Breuil-Kisin-Fargues modules and closelyrelated to Fontaine's $p$-adic Hodge theory over $p$-adic subfields. As anapplication, we characterize basic admissible pairs with complex multiplicationin terms of the transcendence of $p$-adic periods. This generalizes an earlierresult for one-dimensional formal groups and is an unconditional, local,$p$-adic analog of a global, archimedean characterization of CM motives over$\mathbb{C}$ conditional on the standard conjectures, the Hodge conjecture, andthe Grothendieck period conjecture (known unconditionally for abelian varietiesby work Cohen and Shiga and Wolfart).

Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice