Title: Probing BERT in Hyperbolic Spaces. Abstract: Recently, a variety of probing tasks are proposed to discover linguistic properties learned in contextualized word embeddings. Many of these works implicitly assume these embeddings lay in certain metric spaces, typically the Euclidean space. This work considers a family of geometrically special spaces, the hyperbolic spaces, that exhibit better inductive biases for hierarchical structures and may better reveal linguistic hierarchies encoded in contextualized representations. We introduce a $\textit{Poincaré probe}$, a structural probe projecting these embeddings into a Poincaré subspace with explicitly defined hierarchies. We focus on two probing objectives: (a) dependency trees where the hierarchy is defined as head-dependent structures; (b) lexical sentiments where the hierarchy is defined as the polarity of words (positivity and negativity). We argue that a key desideratum of a probe is its sensitivity to the existence of linguistic structures. We apply our probes on BERT, a typical contextualized embedding model. In a syntactic subspace, our probe better recovers tree structures than Euclidean probes, revealing the possibility that the geometry of BERT syntax may not necessarily be Euclidean. In a sentiment subspace, we reveal two possible meta-embeddings for positive and negative sentiments and show how lexically-controlled contextualization would change the geometric localization of embeddings. We demonstrate the findings with our Poincaré probe via extensive experiments and visualization. Our results can be reproduced at https://github.com/FranxYao/PoincareProbe