Let $E$ be an elliptic curve defined over an imaginary quadratic field $F$ of class number $1$. No systematic construction of global points on such an $E$ is known. In this article, we present a $p$-adic analytic construction of points on $E$, which we conjecture to be global, defined over ring class fields of a suitable relative quadratic extension $K/F$. The construction follows ideas of Darmon to produce an analog of Heegner points, which is especially interesting since none of the geometry of modular parametrizations extends to this setting. We present some computational evidence for our construction