For every $$n\ge 2$$, the surface Houghton group $${\mathcal {B}}_n$$is defined as the asymptotically rigid mapping class group of a surface with exactly n ends, all of them non-planar. The groups $${\mathcal {B}}_n$$are analogous to, and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphism of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some $${\mathcal {B}}_n$$. As countable mapping class groups of infinite type surfaces, the groups $$\mathcal {B}n$$lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that $$\mathcal {B}n$$is of type $$\text {F}{n-1}$$, but not of type $$\text {FP}{n}$$, analogous to the braided Houghton groups.