Define the incremental fractional Brownian field with parameter H ∈ (0, 1) by ZH(τ, s) = BH(s +τ)-BH(s), where BH(s) is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). We firstly derive the exact tail asymptotics for the maximum \(M_H^* (T) = \max _{(\tau ,s) \in [a,b] \times [0,T]} Z_H (\tau ,s)/\tau ^H \) of the standardised fractional Brownian motion field, with any fixed 0 0; and we, furthermore, extend the obtained result to the case that T is a positive random variable independent of {BH(s), s ⩾ 0}. As a by-product, we obtain the Gumbel limit law for MH* (T) as T → ∞.