The STAR Collaboration reports a measurement of the transverse single-spin asymmetries, $A_{N}$, for neutral pions produced in polarized proton collisions with protons ($pp$), with aluminum nuclei ($prm{Al}$) and with gold nuclei ($prm{Au}$) at a nucleon-nucleon center-of-mass energy of 200 GeV. Neutral pions are observed in the forward direction relative to the transversely polarized proton beam, in the pseudo-rapidity region $2.7<eta<3.8$. Results are presented for $pi^0$s observed in the STAR FMS electromagnetic calorimeter in narrow Feynman x ($x_F$) and transverse momentum ($p_T$) bins, spanning the range $0.17<x_F<0.81$ and $1.7<p_{T}<6.0$ GeV/$c$. For fixed $x_F<0.47$, the asymmetries are found to rise with increasing transverse momentum. For larger $x_F$, the asymmetry flattens or falls as ${p_T}$ increases. Parametrizing the ratio $r(A) equiv A_N(pA)/A_N(pp)=A^P$ over the kinematic range, the ratio $r(A)$ is found to depend only weakly on $A$, with ${langle}P{rangle} = -0.027 pm 0.005$. No significant difference in $P$ is observed between the low-$p_T$ region, $p_T<2.5$ GeV/$c$, where gluon saturation effects may play a role, and the high-$p_T$ region, $p_T>2.5$ GeV/$c$. It is further observed that the value of $A_N$ is significantly larger for events with a large-$p_T$ isolated $pi^0$ than for events with a non-isolated $pi^0$ accompanied by additional jet-like fragments. The nuclear dependence $r(A)$ is similar for isolated and non-isolated $pi^0$ events.
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