A serious risk to the safety and utility of LLMs is sycophancy, i.e., excessive agreement with and flattery of the user. Yet existing work focuses on only one aspect of sycophancy: agreement with users' explicitly stated beliefs that can be compared to a ground truth. This overlooks forms of sycophancy that arise in ambiguous contexts such as advice and support-seeking, where there is no clear ground truth, yet sycophancy can reinforce harmful implicit assumptions, beliefs, or actions. To address this gap, we introduce a richer theory of social sycophancy in LLMs, characterizing sycophancy as the excessive preservation of a user's face (the positive self-image a person seeks to maintain in an interaction). We present ELEPHANT, a framework for evaluating social sycophancy across five face-preserving behaviors (emotional validation, moral endorsement, indirect language, indirect action, and accepting framing) on two datasets: open-ended questions (OEQ) and Reddit's r/AmITheAsshole (AITA). Across eight models, we show that LLMs consistently exhibit high rates of social sycophancy: on OEQ, they preserve face 47% more than humans, and on AITA, they affirm behavior deemed inappropriate by crowdsourced human judgments in 42% of cases. We further show that social sycophancy is rewarded in preference datasets and is not easily mitigated. Our work provides theoretical grounding and empirical tools (datasets and code) for understanding and addressing this under-recognized but consequential issue.