  We describe {\it JWST}/NIRSpec prism measurements of Ly$\alpha$ emission in
$z\gtrsim 5$ galaxies. We identify Ly$\alpha$ detections in 10 out of 69
galaxies with robust rest-optical emission line redshift measurements at $5\leq
z<7$ in the CEERS and DDT-2750 observations of the EGS field. Galaxies at
$z\simeq 6$ with faint continuum (F150W $=$ 27--29 mag) are found with
extremely large rest-frame Ly$\alpha$ equivalent widths (ranging up to 286 A).
Likely Ly$\alpha$ detections are also seen in two new $z>7$ galaxies ($z=$ 7.49
and 7.17) from the second epoch of CEERS observations, both showing large
Ly$\alpha$ equivalent widths that likely indicate significant transmission
through the IGM. We measure high Ly$\alpha$ escape fractions in the 12
Ly$\alpha$ emitters in our sample (median 0.28), two of which show $f_{\rm
esc}^{ {\rm Ly}\alpha}$ near unity ($>0.80$). We find that $50_{-11}^{+11}$% of
$z\simeq 6$ galaxies with [OIII]+H$\beta$ EW $>$ 1000 A have $f_{\rm esc}^{
{\rm Ly}\alpha}$ $>0.2$, consistent with the fractions found in lower-redshift
samples with matched [OIII]+H$\beta$ EWs. While uncertainties are still
significant, we find that only $10_{-5}^{+9}$% of $z>7$ galaxies with similarly
strong rest optical emission lines show such large $f_{\rm esc}^{ {\rm
Ly}\alpha}$, as may be expected if IGM attenuation of Ly$\alpha$ increases
towards higher redshifts. We identify photometric galaxy overdensities near the
$z\gtrsim 7$ Ly$\alpha$ emitters, potentially providing the ionizing flux
necessary to create large ionized sightlines that facilitate Ly$\alpha$
transmission. Finally, we investigate the absence of Ly$\alpha$ emission in a
comparable (and spectroscopically confirmed) galaxy overdensity at $z=7.88$ in
the Abell 2744 field, discussing new prism spectra of the field obtained with
the UNCOVER program.
