  We construct Lifshitz scalar field theories in 4+1 dimensions which retain
3+1-d Lorentz invariance and therefore ensure a unique limiting speed in the
3+1-d world. Such a construction is potentially useful in developing
field-theoretic ultraviolet completions of extra-dimensional field theories.
The extra dimension y is treated asymmetrically from the usual three spatial
dimensions by introducing derivatives of order 2n with respect to y in the
action. We show that lambda phi^4 theory becomes progressively less
non-renormalisable by power counting as n is increased. This suggests that the
non-local theory obtained in the infinite-n limit may be complete in the
ultraviolet.
