Measurements of the electric resistivity $rho(T)$ under pressure up to 8 GPa were performed on high-quality single-crystals of the Yb-based heavy fermion system $beta$-YbAlB$_4$ in the temperature range $2<T<$ 300 K. In the resistivity data, we observed pressure-induced magnetic ordering above the critical pressure $P_{rm c} sim$ 2 GPa. Clear difference in the phase diagram under pressure using two types of pressure mediums indicates that the transition temperature may be further enhanced under application of uniaxial pressure. With pressure, this phase transition temperature $T_{rm M}$ is enhanced reaching 32 K at 8 GPa, which is the highest transition temperature so far recorded for the Yb-based heavy fermion compounds. The power-law exponent $alpha$ in $rho=rho_0+ AT^{alpha}$ below $T_{rm M}$ gradually changes from 3/2 to 5/2 with increasing pressure from 2 to 8 GPa. In contrast, the resistivity exhibits a $T$-linear behavior in the temperature range 2 $le T le$ 20 K and is insensitive to pressure below $P_{rm c}$. In this pressure regime, the magnetization is also nearly independent of pressure and shows no anomaly above 2 K. Our results indicate that a quantum critical point for $beta$-YbAlB$_4$ is also located near $P_{rm c}$ in addition to the strange metal region near the ambient pressure.
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