  We describe an algorithm to compute the local Coleman-Gross p-adic height at
p on a hyperelliptic curve. Previously, this was only possible using an
algorithm due to Balakrishnan and Besser, which was limited to odd degree.
While we follow their general strategy, our algorithm is significantly faster
and simpler and works for both odd and even degree. We discuss a precision
analysis and an implementation in SageMath. Our work has several applications,
also discussed in this article. These include various versions of the quadratic
Chabauty method, and numerical evidence for a p-adic version of the conjecture
of Birch and Swinnerton-Dyer in cases where this was not previously possible.
