4U~1700--37 is a wind-fed high-mass X-ray binary hosting a compact object, likely a neutron star, accreting from the O6.5~Iaf$^+$ supergiant HD~153919. Despite decades of observations, no coherent pulsations or firm magnetic field estimates exist. We study the hard X-ray timing and broadband spectrum of 4U~1700--37 using \textit{NuSTAR} to search for pulsations and possible cyclotron resonant scattering features. We analysed a ${\sim}$50~ks \textit{NuSTAR} observation from eclipse egress to periastron, performing timing searches with Lomb-- Scargle and $Z^2$ statistics, and modelling the 3--70~keV spectrum with several continua used for accreting pulsars. No coherent pulsations were detected between 100--5000~s, setting 90\% upper limits on the pulsed fraction of 1.5\%--1.8\%. The best-fitting model is a blackbody plus cutoff power law, showing weak absorption-like features near 19.5~keV and 52~keV with ${\sim}$95\% significance. These marginal features may indicate cyclotron lines corresponding to a magnetic field of $B{\approx}1.7\times10^{12}$~G. Within the subsonic settling accretion regime, the expected equilibrium spin period is $P_{\rm eq} {\approx}1.9$~ks. Longer \textit{NuSTAR} exposures are required to confirm these hints.